1358:
2909:
8741:
2345:
3630:
1122:
2599:
2705:
8429:
36:
6287:
987:
3469:
2446:
681:
2193:
2012:
1353:{\displaystyle \operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\mathbf {B} \right)=\operatorname {tr} \left(\mathbf {A} \mathbf {B} ^{\mathsf {T}}\right)=\operatorname {tr} \left(\mathbf {B} ^{\mathsf {T}}\mathbf {A} \right)=\operatorname {tr} \left(\mathbf {B} \mathbf {A} ^{\mathsf {T}}\right)=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ij}\;.}
2904:{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} ),\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} ),\\\operatorname {tr} (\mathbf {A} \mathbf {B} )&=\operatorname {tr} (\mathbf {B} \mathbf {A} ),\end{aligned}}}
6128:
847:
8736:{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\sum _{i=1}^{m}\left(\mathbf {A} \mathbf {B} \right)_{ii}=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ji}=\sum _{j=1}^{n}\sum _{i=1}^{m}b_{ji}a_{ij}=\sum _{j=1}^{n}\left(\mathbf {B} \mathbf {A} \right)_{jj}=\operatorname {tr} (\mathbf {B} \mathbf {A} ).}
466:
7163:
9314:
1670:
8034:
2340:{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} \mathbf {D} )=\operatorname {tr} (\mathbf {B} \mathbf {C} \mathbf {D} \mathbf {A} )=\operatorname {tr} (\mathbf {C} \mathbf {D} \mathbf {A} \mathbf {B} )=\operatorname {tr} (\mathbf {D} \mathbf {A} \mathbf {B} \mathbf {C} ).}
7744:
3625:{\displaystyle {\begin{aligned}\mathbf {P} _{\mathbf {X} }&=\mathbf {X} \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}\\\Longrightarrow \operatorname {tr} \left(\mathbf {P} _{\mathbf {X} }\right)&=\operatorname {rank} (\mathbf {X} ).\end{aligned}}}
1880:
2594:{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )=\operatorname {tr} \left(\left(\mathbf {A} \mathbf {B} \mathbf {C} \right)^{\mathsf {T}}\right)=\operatorname {tr} (\mathbf {C} \mathbf {B} \mathbf {A} )=\operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ),}
8861:
2693:
2432:
3789:
2170:
4308:
5831:
1081:
7037:
5431:
1817:
6282:{\displaystyle B(\mathbf {X} ,\mathbf {Y} )=\operatorname {tr} (\operatorname {ad} (\mathbf {X} )\operatorname {ad} (\mathbf {Y} ))\quad {\text{where }}\operatorname {ad} (\mathbf {X} )\mathbf {Y} ==\mathbf {X} \mathbf {Y} -\mathbf {Y} \mathbf {X} }
5903:
4208:
6841:
826:
9125:
1481:
397:
982:{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} )\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} )\end{aligned}}}
6358:
5633:
7003:
4601:
8754:
6455:
7848:
4403:
676:{\displaystyle \mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}={\begin{pmatrix}1&0&3\\11&5&2\\6&12&-5\end{pmatrix}}}
6100:
2616:
2360:
8928:
7548:
3211:
9522:
which has complement the scalar matrices, and leaves one degree of freedom: any such map is determined by its value on scalars, which is one scalar parameter and hence all are multiple of the trace, a nonzero such
3994:
4228:
9377:
3417:
6693:
6040:
5573:
6555:
9433:
5260:
5737:
8092:
is equivalent to the expressibility of any linear map as the sum of rank-one linear maps. As such, the proof may be written in the notation of tensor products. Then one may consider the multilinear map
3474:
2007:{\displaystyle \operatorname {tr} \left(\mathbf {P} ^{-1}(\mathbf {A} \mathbf {P} )\right)=\operatorname {tr} \left((\mathbf {A} \mathbf {P} )\mathbf {P} ^{-1}\right)=\operatorname {tr} (\mathbf {A} ).}
4102:
9121:
3705:
5757:
2101:
1018:
5341:
2710:
852:
5836:
4139:
2093:
2056:
6758:
688:
4840:
283:
7391:
can be written as the sum of (finitely many) rank-one linear maps. Composing the inverse of the isomorphism with the linear functional obtained above results in a linear functional on
6599:
5679:
5575:
of operators/matrices into traceless operators/matrices and scalars operators/matrices. The projection map onto scalar operators can be expressed in terms of the trace, concretely as:
6299:
9483:
5578:
9030:
5948:
9520:
4875:
9564:
5490:
5294:
5073:
3030:
1758:
1093:
The trace of a square matrix which is the product of two matrices can be rewritten as the sum of entry-wise products of their elements, i.e. as the sum of all elements of their
3096:
6485:
6924:
6753:
6722:
6623:
4543:
1720:
The symmetry of the
Frobenius inner product may be phrased more directly as follows: the matrices in the trace of a product can be switched without changing the result. If
2990:
6919:
6365:
4655:
as a matrix relative to this basis, and taking the trace of this square matrix. The result will not depend on the basis chosen, since different bases will give rise to
3825:
3059:
4330:
9056:
6049:
4980:
3878:
8983:
7158:{\displaystyle \operatorname {tr} (Z)=\operatorname {tr} _{A}\left(\operatorname {tr} _{B}(Z)\right)=\operatorname {tr} _{B}\left(\operatorname {tr} _{A}(Z)\right).}
5019:
6643:
8866:
5980:
4920:
5328:
3845:
3116:
3010:
2933:
9309:{\displaystyle f(\mathbf {A} )=\sum _{i,j}_{ij}f\left(e_{ij}\right)=\sum _{i}_{ii}f\left(e_{11}\right)=f\left(e_{11}\right)\operatorname {tr} (\mathbf {A} ).}
3338:
Conversely, any square matrix with zero trace is a linear combination of the commutators of pairs of matrices. Moreover, any square matrix with zero trace is
1665:{\displaystyle 0\leq \left^{2}\leq \operatorname {tr} \left(\mathbf {A} ^{2}\right)\operatorname {tr} \left(\mathbf {B} ^{2}\right)\leq \left^{2}\left^{2}\ ,}
8363:
2601:
where the first equality is because the traces of a matrix and its transpose are equal. Note that this is not true in general for more than three factors.
8029:{\displaystyle \operatorname {tr} (S\circ T)=\sum _{i}\sum _{j}\psi _{j}(v_{i})\varphi _{i}(w_{j})=\sum _{j}\sum _{i}\varphi _{i}(w_{j})\psi _{j}(v_{i}).}
6046:
th root times scalars, and this does not in general define a function, so the determinant does not split and the general linear group does not decompose:
9319:
3373:
7739:{\displaystyle (S\circ T)(u)=\sum _{i}\varphi _{i}\left(\sum _{j}\psi _{j}(u)w_{j}\right)v_{i}=\sum _{i}\sum _{j}\psi _{j}(u)\varphi _{i}(w_{j})v_{i}}
6515:
3148:
9676:
5338:
is
Abelian (the Lie bracket vanishes), the fact that this is a map of Lie algebras is exactly the statement that the trace of a bracket vanishes:
53:
4050:
3335:). In particular, using similarity invariance, it follows that the identity matrix is never similar to the commutator of any pair of matrices.
8856:{\displaystyle \mathbf {A} ={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}0&0\\1&0\end{pmatrix}},}
10016:
3942:
3245:
is a real matrix and some (or all) of the eigenvalues are complex numbers. This may be regarded as a consequence of the existence of the
2688:{\displaystyle \operatorname {tr} (\mathbf {A} \otimes \mathbf {B} )=\operatorname {tr} (\mathbf {A} )\operatorname {tr} (\mathbf {B} ).}
10052:
2427:{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )\neq \operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ).}
6648:
5985:
5518:
9887:
9382:
9935:
9776:
9751:
9697:
4323:
From this (or from the connection between the trace and the eigenvalues), one can derive a relation between the trace function, the
119:
8403:
5219:
100:
5688:
72:
7350:. The universal property of the tensor product, just as used previously, says that this bilinear map is induced by a linear map
9061:
3784:{\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}^{k}\right)=\operatorname {tr} \left(\mathbf {I} _{n}\right)=n\equiv 0}
8373:
2165:{\displaystyle \operatorname {tr} \left(\mathbf {b} \mathbf {a} ^{\textsf {T}}\right)=\mathbf {a} ^{\textsf {T}}\mathbf {b} }
57:
4303:{\displaystyle d\det(\mathbf {A} )=\operatorname {tr} {\big (}\operatorname {adj} (\mathbf {A} )\cdot d\mathbf {A} {\big )}}
1475:
79:
10080:
9915:
1085:
This follows immediately from the fact that transposing a square matrix does not affect elements along the main diagonal.
6868:
5500:
of matrices with determinant 1. The special linear group consists of the matrices which do not change volume, while the
4313:
10075:
5205:
3463:
2061:
2024:
1377:
1094:
86:
10008:
9978:
5512:
5501:
4804:
4629:
4107:
7219:. Such a trace is not uniquely defined; it can always at least be modified by multiplication by a nonzero scalar.
46:
3881:
3439:
68:
8276:. The established symmetry upon composition with the trace map then establishes the equality of the two traces.
7168:
6562:
5644:
3421:
1685:
10070:
5126:
5102:
9570:
and thus every element in it is a linear combination of commutators of some pairs of elements, otherwise the
9438:
7418:
is straightforward to prove, and was given above. In the present perspective, one is considering linear maps
5826:{\displaystyle 0\to {\mathfrak {sl}}_{n}\to {\mathfrak {gl}}_{n}{\overset {\operatorname {tr} }{\to }}K\to 0}
1076:{\displaystyle \operatorname {tr} (\mathbf {A} )=\operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\right).}
10097:
8988:
8074:
The above proof can be regarded as being based upon tensor products, given that the fundamental identity of
5908:
5426:{\displaystyle \operatorname {tr} ()=0{\text{ for each }}\mathbf {A} ,\mathbf {B} \in {\mathfrak {gl}}_{n}.}
5133:
5076:
3884:, possibly changed of sign, according to the convention in the definition of the characteristic polynomial.
1700:
1447:
9488:
4845:
1812:{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\operatorname {tr} (\mathbf {B} \mathbf {A} )}
10102:
9567:
9535:
5461:
5265:
5024:
4644:
3933:
5898:{\displaystyle 1\to \operatorname {SL} _{n}\to \operatorname {GL} _{n}{\overset {\det }{\to }}K^{*}\to 1}
4203:{\displaystyle \det(\mathbf {I} +\mathbf {\Delta A} )\approx 1+\operatorname {tr} (\mathbf {\Delta A} ).}
3015:
10107:
10000:
6836:{\displaystyle \phi (\mathbf {X} ,\mathbf {Y} )={\text{tr}}_{V}(\rho (\mathbf {X} )\rho (\mathbf {Y} ))}
5682:
4636:
3659:
3246:
2015:
1847:
9833:"Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix"
3064:
821:{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{3}a_{ii}=a_{11}+a_{22}+a_{33}=1+5+(-5)=1}
6463:
6727:
6698:
6604:
5751:
5497:
5137:
1704:
1002:
8383:
7179:
4663:
4221:
3332:
443:
392:{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}a_{ii}=a_{11}+a_{22}+\dots +a_{nn}}
247:
2942:
93:
9860:
8398:
7377:. This fundamental fact is a straightforward consequence of the existence of a (finite) basis of
7374:
7234:
6884:
6353:{\displaystyle (\mathbf {X} ,\mathbf {Y} )\mapsto \operatorname {tr} (\mathbf {X} \mathbf {Y} ).}
5456:
4471:
4324:
3435:
1467:
239:
into itself, since all matrices describing such an operator with respect to a basis are similar.
9882:. Graduate Studies in Mathematics. Vol. 157 (2nd ed.). American Mathematical Society.
5628:{\displaystyle \mathbf {A} \mapsto {\frac {1}{n}}\operatorname {tr} (\mathbf {A} )\mathbf {I} .}
4659:, allowing for the possibility of a basis-independent definition for the trace of a linear map.
9793:
7012:
is another generalization of the trace that is operator-valued. The trace of a linear operator
6493:), every such bilinear form is proportional to each other; in particular, to the Killing form.
4106:
Everything in the present section applies as well to any square matrix with coefficients in an
3804:
3038:
10048:
10012:
9931:
9883:
9852:
9813:
9772:
9747:
9693:
9681:
9667:
9636:
8342:
6879:
4656:
4014:
3645:
2936:
2610:
1714:
228:
9794:"A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines"
9035:
4925:
3850:
10044:
9966:
9949:
9923:
9844:
9805:
9711:
9685:
8958:
8358:
6856:
4989:
3666:
3428:
3339:
2440:
10026:
9986:
9945:
9707:
6628:
10022:
9982:
9953:
9941:
9919:
9715:
9703:
9571:
8346:
5091:
4317:
4131:
3362:
1689:
841:
439:
232:
5957:
5109:
equal to one. Then, if the square of the trace is 4, the corresponding transformation is
4880:
2014:
Similarity invariance is the crucial property of the trace in order to discuss traces of
6998:{\displaystyle \operatorname {tr} (K)=\sum _{n}\left\langle e_{n},Ke_{n}\right\rangle ,}
10036:
9595:
8423:
8393:
8378:
7284:
7280:
6864:
5313:
5098:
4006:
3903:
3830:
3101:
2995:
2918:
2183:
1471:
193:
133:
4596:{\displaystyle d\operatorname {tr} (\mathbf {X} )=\operatorname {tr} (d\mathbf {X} ).}
10091:
7009:
6860:
6110:
4744:; the trace of a general element is defined by linearity. The trace of a linear map
1432:
272:
158:
141:
6450:{\displaystyle \operatorname {tr} (\mathbf {X} )=\operatorname {tr} (\mathbf {Z} ).}
9864:
7227:
6290:
4632:
4502:
4408:
3316:
1463:
236:
7426:, and viewing them as sums of rank-one maps, so that there are linear functionals
7406:
Using the definition of trace as the sum of diagonal elements, the matrix formula
7296:
automatically implies that this bilinear map is induced by a linear functional on
9671:
4443:. The components of this vector field are linear functions (given by the rows of
4398:{\displaystyle \det(\exp(\mathbf {A} ))=\exp(\operatorname {tr} (\mathbf {A} )).}
17:
9974:
8388:
8368:
6852:
6095:{\displaystyle \operatorname {GL} _{n}\neq \operatorname {SL} _{n}\times K^{*}.}
5493:
5106:
4784:, one can show that this gives the same definition of the trace as given above.
4316:
of the determinant at an arbitrary square matrix, in terms of the trace and the
4217:
4034:
4010:
3899:
1393:
435:
243:
35:
3671:
When the characteristic of the base field is zero, the converse also holds: if
9996:
9927:
9809:
7223:
5436:
4697:
4450:
4213:
3923:
3331:
from operators to scalars", as the commutator of scalars is trivial (it is an
3280:
3230:
1693:
200:
9856:
9817:
8923:{\displaystyle \mathbf {AB} ={\begin{pmatrix}1&0\\0&0\end{pmatrix}},}
4773:
under the above mentioned canonical isomorphism. Using an explicit basis for
3206:{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}\lambda _{i}}
9848:
5951:
1012:
1688:
of the same size. The
Frobenius inner product and norm arise frequently in
1474:, and it satisfies a submultiplicative property, as can be proven with the
1407:
is a sum of squares and hence is nonnegative, equal to zero if and only if
9832:
9689:
1431:. These demonstrate the positive-definiteness and symmetry required of an
6362:
The form is symmetric, non-degenerate and associative in the sense that:
4758:
can then be defined as the trace, in the above sense, of the element of
3249:, together with the similarity-invariance of the trace discussed above.
1877:
of the same dimensions, is a fundamental consequence. This is proved by
5082:
More sophisticated stochastic estimators of trace have been developed.
4540:
The trace is a linear operator, hence it commutes with the derivative:
3989:{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i}\lambda _{i}}
5638:
4635:), we can define the trace of this map by considering the trace of a
4530:
2095:, the trace of the outer product is equivalent to the inner product:
231:
have the same trace. As a consequence one can define the trace of a
7167:
For more properties and a generalization of the partial trace, see
9372:{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus k,}
5121:. Finally, if the square is greater than 4, the transformation is
3412:{\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}\right)=n}
2912:
6688:{\displaystyle \rho :{\mathfrak {g}}\rightarrow {\text{End}}(V).}
6035:{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K}
5568:{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K}
6550:{\displaystyle \operatorname {tr} (\mathbf {X} \mathbf {Y} )=0.}
9428:{\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA)}
9637:"Rank, trace, determinant, transpose, and inverse of matrices"
4797:
The trace can be estimated unbiasedly by "Hutchinson's trick":
199:
It can be proven that the trace of a matrix is the sum of its
29:
5255:{\displaystyle \operatorname {tr} :{\mathfrak {gl}}_{n}\to K}
5732:{\displaystyle {\mathfrak {gl}}_{n}\to {\mathfrak {gl}}_{n}}
3442:
of the corresponding permutation, because the diagonal term
2443:
matrices are considered, any permutation is allowed, since:
1834:, and also since the trace of either does not usually equal
8047:
reversed, one finds exactly the same formula, proving that
7403:. This linear functional is exactly the same as the trace.
5204:
The trace also plays a central role in the distribution of
4407:
A related characterization of the trace applies to linear
4097:{\displaystyle \det(\mathbf {A} )=\prod _{i}\lambda _{i}.}
3342:
to a square matrix with diagonal consisting of all zeros.
203:(counted with multiplicities). It can also be proven that
9798:
Communications in
Statistics - Simulation and Computation
9116:{\displaystyle f\left(e_{jj}\right)=f\left(e_{11}\right)}
8300:; in the language of linear maps, it assigns to a scalar
8250:. It can be seen that this coincides with the linear map
8176:, and this is unchanged if one were to have started with
8157:. Further composition with the trace map then results in
3898:
is a linear operator represented by a square matrix with
7381:, and can also be phrased as saying that any linear map
7005:
and is finite and independent of the orthonormal basis.
6293:, which is used for the classification of Lie algebras.
3315:
is linear. One can state this as "the trace is a map of
9316:
More abstractly, this corresponds to the decomposition
6851:
The concept of trace of a matrix is generalized to the
6042:, but the splitting of the determinant would be as the
5090:
If a 2 x 2 real matrix has zero trace, its square is a
3431:
is real, because the elements on the diagonal are real.
8886:
8819:
8771:
6559:
There is a generalization to a general representation
1707:
of all complex matrices of a fixed size, by replacing
612:
483:
192:. This is a misnomer, but widely used, such as in the
9538:
9491:
9441:
9385:
9322:
9128:
9064:
9038:
8991:
8961:
8869:
8757:
8432:
7851:
7551:
7040:
6927:
6887:
6761:
6730:
6701:
6651:
6631:
6607:
6565:
6518:
6466:
6368:
6302:
6131:
6052:
5988:
5960:
5911:
5839:
5760:
5747:
makes this a projection, yielding the formula above.
5691:
5647:
5581:
5521:
5464:
5344:
5316:
5268:
5222:
5027:
4992:
4928:
4883:
4848:
4807:
4546:
4333:
4231:
4142:
4053:
3945:
3853:
3833:
3807:
3708:
3472:
3376:
3151:
3104:
3067:
3041:
3018:
2998:
2945:
2921:
2708:
2619:
2449:
2363:
2196:
2104:
2064:
2027:
1883:
1761:
1484:
1413:
is zero. Furthermore, as noted in the above formula,
1125:
1021:
850:
691:
469:
286:
2357:
Arbitrary permutations are not allowed: in general,
449:. The trace is not defined for non-square matrices.
9742:Lipschutz, Seymour; Lipson, Marc (September 2005).
7226:is the generalization of a trace to the setting of
3239:counted with multiplicity. This holds true even if
60:. Unsourced material may be challenged and removed.
9558:
9514:
9477:
9427:
9371:
9308:
9115:
9050:
9024:
8977:
8922:
8855:
8735:
8028:
7738:
7157:
6997:
6913:
6835:
6747:
6716:
6687:
6637:
6617:
6593:
6549:
6479:
6449:
6352:
6281:
6094:
6034:
5974:
5942:
5897:
5825:
5731:
5673:
5627:
5567:
5484:
5425:
5322:
5288:
5254:
5067:
5013:
4974:
4914:
4869:
4834:
4595:
4397:
4302:
4202:
4096:
3988:
3872:
3839:
3819:
3783:
3624:
3411:
3205:
3110:
3090:
3053:
3024:
3004:
2984:
2927:
2903:
2687:
2593:
2426:
2339:
2164:
2087:
2050:
2006:
1811:
1664:
1352:
1075:
981:
820:
675:
391:
9737:
9735:
9733:
9731:
8359:Trace of a tensor with respect to a metric tensor
7373:is finite-dimensional, then this linear map is a
1699:The Frobenius inner product may be extended to a
167:. The trace is only defined for a square matrix (
8341:. These structures can be axiomatized to define
8196:instead. One may also consider the bilinear map
5874:
5435:The kernel of this map, a matrix whose trace is
4459:is a constant function, whose value is equal to
4334:
4235:
4143:
4054:
2613:of two matrices is the product of their traces:
2088:{\displaystyle \mathbf {b} \in \mathbb {R} ^{n}}
2051:{\displaystyle \mathbf {a} \in \mathbb {R} ^{n}}
4723:. Then the trace of the indecomposable element
4485:represents the velocity of a fluid at location
4474:, one can interpret this in terms of flows: if
2939:on the space of square matrices that satisfies
5105:. First, the matrix is normalized to make its
1470:derived from this inner product is called the
1466:of all real matrices of fixed dimensions. The
242:The trace is related to the derivative of the
157:, is defined to be the sum of elements on the
8422:This is immediate from the definition of the
6843:is symmetric and invariant due to cyclicity.
5504:is the matrices which do not alter volume of
4835:{\displaystyle W\in \mathbb {R} ^{n\times n}}
4295:
4260:
3797:Relationship to the characteristic polynomial
2915:a scalar multiple in the following sense: If
1752:real or complex matrices, respectively, then
8:
9771:(2nd ed.). Cambridge University Press.
9767:Horn, Roger A.; Johnson, Charles R. (2013).
7237:generalizes the trace to arbitrary tensors.
5937:
5931:
5056:
5028:
3279:matrices, the trace of the (ring-theoretic)
161:(from the upper left to the lower right) of
7309:Similarly, there is a natural bilinear map
5954:. However, the trace splits naturally (via
4986:Usually, the random vector is sampled from
4982:. (Proof: expand the expectation directly.)
4212:Precisely this means that the trace is the
6459:For a complex simple Lie algebra (such as
4124:is a square matrix with small entries and
1346:
9905:
9903:
9901:
9899:
9880:Mathematical Methods in Quantum Mechanics
9831:Avron, Haim; Toledo, Sivan (2011-04-11).
9550:
9541:
9540:
9537:
9503:
9494:
9493:
9490:
9440:
9384:
9354:
9345:
9344:
9334:
9325:
9324:
9321:
9295:
9276:
9252:
9232:
9223:
9214:
9194:
9174:
9165:
9150:
9135:
9127:
9103:
9076:
9063:
9037:
9003:
8990:
8966:
8960:
8881:
8870:
8868:
8814:
8806:
8766:
8758:
8756:
8722:
8717:
8696:
8686:
8681:
8669:
8658:
8642:
8629:
8619:
8608:
8598:
8587:
8571:
8558:
8548:
8537:
8527:
8516:
8500:
8490:
8485:
8473:
8462:
8447:
8442:
8431:
8014:
8001:
7988:
7975:
7965:
7955:
7939:
7926:
7913:
7900:
7890:
7880:
7850:
7730:
7717:
7704:
7685:
7675:
7665:
7652:
7637:
7618:
7608:
7593:
7583:
7550:
7241:Traces in the language of tensor products
7129:
7111:
7081:
7063:
7039:
6981:
6965:
6950:
6926:
6905:
6895:
6886:
6822:
6808:
6793:
6788:
6776:
6768:
6760:
6731:
6729:
6708:
6703:
6700:
6668:
6659:
6658:
6650:
6630:
6609:
6608:
6606:
6594:{\displaystyle (\rho ,{\mathfrak {g}},V)}
6576:
6575:
6564:
6533:
6528:
6517:
6468:
6467:
6465:
6436:
6428:
6420:
6394:
6386:
6378:
6367:
6339:
6334:
6314:
6306:
6301:
6274:
6269:
6261:
6256:
6245:
6237:
6226:
6218:
6204:
6192:
6175:
6146:
6138:
6130:
6083:
6070:
6057:
6051:
6020:
6011:
6010:
6000:
5991:
5990:
5987:
5964:
5959:
5916:
5910:
5883:
5869:
5863:
5850:
5838:
5804:
5798:
5789:
5788:
5778:
5769:
5768:
5759:
5739:mapping onto scalars, and multiplying by
5723:
5714:
5713:
5703:
5694:
5693:
5690:
5674:{\displaystyle K\to {\mathfrak {gl}}_{n}}
5665:
5656:
5655:
5646:
5637:Formally, one can compose the trace (the
5617:
5609:
5590:
5582:
5580:
5553:
5544:
5543:
5533:
5524:
5523:
5520:
5476:
5467:
5466:
5463:
5414:
5405:
5404:
5395:
5387:
5382:
5365:
5357:
5343:
5315:
5280:
5271:
5270:
5267:
5240:
5231:
5230:
5221:
5167:are equivalent (up to change of basis on
5059:
5045:
5038:
5026:
4991:
4939:
4927:
4897:
4882:
4861:
4857:
4856:
4847:
4820:
4816:
4815:
4806:
4662:Such a definition can be given using the
4582:
4559:
4545:
4381:
4349:
4332:
4294:
4293:
4288:
4274:
4259:
4258:
4241:
4230:
4186:
4157:
4149:
4141:
4085:
4075:
4060:
4052:
3980:
3970:
3955:
3944:
3858:
3852:
3832:
3806:
3759:
3754:
3730:
3725:
3720:
3707:
3607:
3580:
3579:
3574:
3549:
3548:
3543:
3533:
3523:
3516:
3515:
3510:
3498:
3484:
3483:
3478:
3473:
3471:
3393:
3388:
3375:
3197:
3187:
3176:
3161:
3150:
3103:
3074:
3066:
3040:
3017:
2997:
2944:
2920:
2886:
2881:
2857:
2852:
2828:
2801:
2774:
2754:
2730:
2722:
2709:
2707:
2674:
2657:
2637:
2629:
2618:
2580:
2575:
2570:
2550:
2545:
2540:
2517:
2516:
2506:
2501:
2496:
2469:
2464:
2459:
2448:
2413:
2408:
2403:
2383:
2378:
2373:
2362:
2326:
2321:
2316:
2311:
2291:
2286:
2281:
2276:
2256:
2251:
2246:
2241:
2221:
2216:
2211:
2206:
2195:
2157:
2151:
2150:
2149:
2144:
2129:
2128:
2127:
2122:
2116:
2103:
2079:
2075:
2074:
2065:
2063:
2042:
2038:
2037:
2028:
2026:
1993:
1967:
1962:
1953:
1948:
1918:
1913:
1901:
1896:
1882:
1801:
1796:
1776:
1771:
1760:
1650:
1637:
1616:
1603:
1575:
1570:
1549:
1544:
1524:
1511:
1506:
1483:
1462:. This is a natural inner product on the
1337:
1324:
1314:
1303:
1293:
1282:
1263:
1262:
1257:
1251:
1227:
1220:
1219:
1214:
1187:
1186:
1181:
1175:
1151:
1144:
1143:
1138:
1124:
1059:
1058:
1053:
1031:
1020:
967:
940:
916:
896:
872:
864:
851:
849:
779:
766:
753:
737:
727:
716:
701:
690:
607:
590:
578:
566:
552:
540:
528:
514:
502:
490:
478:
470:
468:
380:
361:
348:
332:
322:
311:
296:
285:
120:Learn how and when to remove this message
8279:For any finite dimensional vector space
8234:, which is then induced by a linear map
6878:is a trace-class operator, then for any
5127:classification of Möbius transformations
3422:generalizations of dimension using trace
227:of appropriate sizes. This implies that
9677:CRC Concise Encyclopedia of Mathematics
9619:
9478:{\displaystyle \operatorname {tr} ()=0}
8415:
6755:is defined as above. The bilinear form
5928:
4033:on the main diagonal. In contrast, the
1822:This is notable both for the fact that
9025:{\displaystyle f\left(e_{ij}\right)=0}
5943:{\displaystyle K^{*}=K\setminus \{0\}}
3550:
3517:
3466:is the dimension of the target space.
3365:is the dimension of the space, namely
2518:
2021:Additionally, for real column vectors
1264:
1221:
1188:
1145:
1060:
9744:Theory and Problems of Linear Algebra
9515:{\displaystyle {\mathfrak {sl}}_{n},}
4870:{\displaystyle u\in \mathbb {R} ^{n}}
4777:and the corresponding dual basis for
3061:matrices, imposing the normalization
1396:. According to the above expression,
7:
9662:
9660:
9658:
9656:
9631:
9629:
9627:
9625:
9623:
9559:{\displaystyle {\mathfrak {sl}}_{n}}
7026:is equal to the partial traces over
5485:{\displaystyle {\mathfrak {sl}}_{n}}
5289:{\displaystyle {\mathfrak {gl}}_{n}}
5068:{\displaystyle \{\pm n^{-1/2}\}^{n}}
3791:, but the identity is not nilpotent.
58:adding citations to reliable sources
27:Sum of elements on the main diagonal
10041:Linear Algebra and its Applications
9545:
9542:
9498:
9495:
9349:
9346:
9329:
9326:
6660:
6610:
6577:
6472:
6469:
6296:The trace defines a bilinear form:
6015:
6012:
5995:
5992:
5793:
5790:
5773:
5770:
5718:
5715:
5698:
5695:
5660:
5657:
5548:
5545:
5528:
5525:
5471:
5468:
5409:
5406:
5275:
5272:
5235:
5232:
5216:The trace is a map of Lie algebras
5113:. If the square is in the interval
3702:dimensions is a counterexample, as
3346:Traces of special kinds of matrices
3025:{\displaystyle \operatorname {tr} }
8039:Following the same procedure with
7249:, there is a natural bilinear map
6645:is a homomorphism of Lie algebras
4610:In general, given some linear map
4312:is more general and describes the
3459:th point is fixed and 0 otherwise.
442:, or more generally elements of a
188:= 0 then the matrix is said to be
180:In mathematical physics texts, if
25:
9792:Hutchinson, M.F. (January 1989).
9746:. Schaum's Outline. McGraw-Hill.
8985:the standard basis and note that
4220:function at the identity matrix.
3644:More generally, the trace of any
3091:{\displaystyle f(\mathbf {I} )=n}
9680:(2nd ed.). Boca Raton, FL:
9583:This follows from the fact that
9296:
9224:
9166:
9136:
8874:
8871:
8807:
8759:
8723:
8718:
8687:
8682:
8491:
8486:
8448:
8443:
8283:, there is a natural linear map
6823:
6809:
6777:
6769:
6534:
6529:
6480:{\displaystyle {\mathfrak {sl}}}
6437:
6429:
6421:
6395:
6387:
6379:
6340:
6335:
6315:
6307:
6275:
6270:
6262:
6257:
6246:
6238:
6227:
6219:
6193:
6176:
6147:
6139:
5618:
5610:
5583:
5396:
5388:
5366:
5358:
4583:
4560:
4382:
4350:
4289:
4275:
4242:
4190:
4187:
4161:
4158:
4150:
4061:
3999:This follows from the fact that
3956:
3755:
3721:
3608:
3581:
3575:
3544:
3524:
3511:
3499:
3485:
3479:
3389:
3162:
3075:
2887:
2882:
2858:
2853:
2829:
2802:
2775:
2755:
2731:
2723:
2702:The following three properties:
2675:
2658:
2638:
2630:
2581:
2576:
2571:
2551:
2546:
2541:
2507:
2502:
2497:
2470:
2465:
2460:
2414:
2409:
2404:
2384:
2379:
2374:
2327:
2322:
2317:
2312:
2292:
2287:
2282:
2277:
2257:
2252:
2247:
2242:
2222:
2217:
2212:
2207:
2158:
2145:
2123:
2117:
2066:
2029:
1994:
1963:
1954:
1949:
1919:
1914:
1897:
1802:
1797:
1777:
1772:
1638:
1604:
1571:
1545:
1512:
1507:
1258:
1252:
1228:
1215:
1182:
1176:
1152:
1139:
1054:
1032:
968:
941:
917:
897:
873:
865:
702:
471:
297:
34:
10007:(2nd ed.). Cambridge, UK:
9674:. In Weisstein, Eric W. (ed.).
8805:
7201:which vanishes on commutators;
7190:is often defined to be any map
7016:which lives on a product space
6748:{\displaystyle {\text{End}}(V)}
6717:{\displaystyle {\text{tr}}_{V}}
6618:{\displaystyle {\mathfrak {g}}}
6203:
5097:The trace of a 2 × 2
3122:Trace as the sum of eigenvalues
1686:positive semi-definite matrices
45:needs additional citations for
9878:Teschl, G. (30 October 2014).
9466:
9463:
9451:
9448:
9422:
9413:
9401:
9392:
9300:
9292:
9229:
9220:
9171:
9162:
9140:
9132:
8727:
8714:
8452:
8439:
8404:von Neumann's trace inequality
8020:
8007:
7994:
7981:
7945:
7932:
7919:
7906:
7870:
7858:
7723:
7710:
7697:
7691:
7630:
7624:
7573:
7567:
7564:
7552:
7144:
7138:
7096:
7090:
7053:
7047:
6940:
6934:
6902:
6888:
6830:
6827:
6819:
6813:
6805:
6799:
6781:
6765:
6742:
6736:
6679:
6673:
6665:
6588:
6566:
6538:
6525:
6441:
6433:
6417:
6414:
6402:
6399:
6383:
6375:
6344:
6331:
6322:
6319:
6303:
6250:
6234:
6223:
6215:
6200:
6197:
6189:
6180:
6172:
6163:
6151:
6135:
5889:
5871:
5856:
5843:
5817:
5806:
5784:
5764:
5709:
5651:
5614:
5606:
5587:
5511:In fact, there is an internal
5455:, and these matrices form the
5373:
5370:
5354:
5351:
5246:
5008:
4996:
4969:
4963:
4951:
4932:
4903:
4887:
4587:
4576:
4564:
4556:
4389:
4386:
4378:
4369:
4357:
4354:
4346:
4337:
4327:function, and the determinant:
4279:
4271:
4246:
4238:
4194:
4183:
4165:
4146:
4065:
4057:
3960:
3952:
3612:
3604:
3560:
3166:
3158:
3079:
3071:
2976:
2967:
2958:
2949:
2891:
2878:
2862:
2849:
2833:
2825:
2806:
2795:
2779:
2771:
2759:
2751:
2735:
2719:
2679:
2671:
2662:
2654:
2642:
2626:
2585:
2567:
2555:
2537:
2474:
2456:
2418:
2400:
2388:
2370:
2331:
2308:
2296:
2273:
2261:
2238:
2226:
2203:
1998:
1990:
1958:
1945:
1923:
1910:
1806:
1793:
1781:
1768:
1642:
1634:
1608:
1600:
1516:
1503:
1380:) then the above operation on
1036:
1028:
972:
964:
945:
934:
921:
913:
901:
893:
877:
861:
809:
800:
706:
698:
301:
293:
1:
9916:Graduate Texts in Mathematics
4134:, then we have approximately
4047:of its eigenvalues; that is,
3698:is positive, the identity in
2698:Characterization of the trace
2180:More generally, the trace is
1372:matrix as a vector of length
235:mapping a finite-dimensional
5132:The trace is used to define
2985:{\displaystyle f(xy)=f(yx),}
2605:Trace of a Kronecker product
1392:coincides with the standard
10076:Encyclopedia of Mathematics
9918:. Vol. 155. New York:
8345:in the abstract setting of
8316:. Sometimes this is called
6914:{\displaystyle (e_{n})_{n}}
3932:(listed according to their
3888:Relationship to eigenvalues
1850:of the trace, meaning that
69:"Trace" linear algebra
10124:
10071:"Trace of a square matrix"
10009:Cambridge University Press
9979:Chelsea Publishing Company
9910:Kassel, Christian (1995).
8374:Golden–Thompson inequality
7757:. The rank-one linear map
7169:traced monoidal categories
5502:special linear Lie algebra
5296:of linear operators on an
4606:Trace of a linear operator
4108:algebraically closed field
1871:and any invertible matrix
9928:10.1007/978-1-4612-0783-2
9810:10.1080/03610918908812806
5310:matrices with entries in
5021:(normal distribution) or
3882:characteristic polynomial
3820:{\displaystyle n\times n}
3054:{\displaystyle n\times n}
1476:Cauchy–Schwarz inequality
408:denotes the entry on the
9574:would be a proper ideal.
6921:, the trace is given by
6863:, and the analog of the
4417:, define a vector field
4114:Derivative relationships
3934:algebraic multiplicities
3691:When the characteristic
2436:However, if products of
989:for all square matrices
9849:10.1145/1944345.1944349
9485:) defines the trace on
9051:{\displaystyle i\neq j}
8364:Characteristic function
5641:map) with the unit map
5077:Rademacher distribution
4975:{\displaystyle E=tr(W)}
3873:{\displaystyle t^{n-1}}
2911:characterize the trace
1828:does not usually equal
1701:hermitian inner product
1448:Frobenius inner product
1435:; it is common to call
1097:. Phrased directly, if
9971:The Theory of Matrices
9568:semisimple Lie algebra
9560:
9516:
9479:
9429:
9373:
9310:
9117:
9052:
9026:
8979:
8978:{\displaystyle e_{ij}}
8924:
8857:
8737:
8674:
8624:
8603:
8553:
8532:
8478:
8030:
7740:
7159:
6999:
6915:
6837:
6749:
6718:
6689:
6639:
6619:
6595:
6551:
6481:
6451:
6354:
6283:
6096:
6036:
5976:
5944:
5899:
5833:which is analogous to
5827:
5733:
5675:
5629:
5569:
5486:
5439:, is often said to be
5427:
5324:
5290:
5256:
5140:. Two representations
5103:Möbius transformations
5069:
5015:
5014:{\displaystyle N(0,I)}
4984:
4976:
4916:
4871:
4836:
4643:, that is, choosing a
4597:
4399:
4304:
4204:
4098:
3990:
3874:
3847:is the coefficient of
3841:
3821:
3785:
3626:
3413:
3207:
3192:
3112:
3092:
3055:
3026:
3006:
2986:
2929:
2905:
2689:
2595:
2428:
2341:
2166:
2089:
2052:
2016:linear transformations
2008:
1865:for any square matrix
1813:
1666:
1362:If one views any real
1354:
1319:
1298:
1077:
983:
822:
732:
677:
393:
327:
9690:10.1201/9781420035223
9561:
9517:
9480:
9430:
9374:
9311:
9118:
9053:
9027:
8980:
8925:
8858:
8738:
8654:
8604:
8583:
8533:
8512:
8458:
8031:
7741:
7245:Given a vector space
7160:
7000:
6916:
6838:
6750:
6719:
6690:
6640:
6638:{\displaystyle \rho }
6620:
6596:
6552:
6482:
6452:
6355:
6284:
6125:are square matrices)
6097:
6037:
5977:
5945:
5900:
5828:
5752:short exact sequences
5734:
5676:
5630:
5570:
5487:
5428:
5330:) to the Lie algebra
5325:
5291:
5262:from the Lie algebra
5257:
5138:group representations
5070:
5016:
4977:
4917:
4872:
4837:
4799:
4664:canonical isomorphism
4637:matrix representation
4598:
4400:
4305:
4205:
4099:
3991:
3875:
3842:
3822:
3786:
3627:
3414:
3247:Jordan canonical form
3208:
3172:
3113:
3093:
3056:
3027:
3007:
2987:
2930:
2906:
2690:
2596:
2429:
2350:This is known as the
2342:
2167:
2090:
2053:
2009:
1848:similarity-invariance
1814:
1667:
1376:(an operation called
1355:
1299:
1278:
1078:
1015:have the same trace:
984:
823:
712:
678:
394:
307:
215:for any two matrices
9536:
9489:
9439:
9383:
9320:
9126:
9062:
9036:
8989:
8959:
8867:
8863:then the product is
8755:
8430:
7849:
7549:
7448:and nonzero vectors
7038:
6925:
6885:
6759:
6728:
6699:
6649:
6629:
6605:
6563:
6516:
6464:
6366:
6300:
6129:
6050:
5986:
5958:
5909:
5837:
5758:
5689:
5645:
5579:
5519:
5498:special linear group
5462:
5384: for each
5342:
5314:
5300:-dimensional space (
5266:
5220:
5101:is used to classify
5025:
4990:
4926:
4881:
4846:
4805:
4793:Stochastic estimator
4788:Numerical algorithms
4544:
4505:of the fluid out of
4331:
4229:
4140:
4051:
3943:
3851:
3831:
3805:
3706:
3470:
3374:
3340:unitarily equivalent
3149:
3118:equal to the trace.
3102:
3065:
3039:
3016:
2996:
2943:
2919:
2706:
2617:
2447:
2361:
2194:
2102:
2062:
2025:
1881:
1759:
1705:complex vector space
1482:
1123:
1019:
848:
689:
467:
284:
54:improve this article
9641:fourier.eng.hmc.edu
8936:) = 1 ≠ 0 ⋅ 0 = tr(
8930:and the traces are
8224:to the composition
7180:associative algebra
5975:{\displaystyle 1/n}
4915:{\displaystyle E=I}
3735:
3333:Abelian Lie algebra
3253:Trace of commutator
9837:Journal of the ACM
9682:Chapman & Hall
9668:Weisstein, Eric W.
9556:
9512:
9475:
9425:
9369:
9306:
9219:
9161:
9113:
9048:
9022:
8975:
8920:
8911:
8853:
8844:
8796:
8733:
8399:Trace inequalities
8343:categorical traces
8026:
7970:
7960:
7895:
7885:
7736:
7680:
7670:
7613:
7588:
7375:linear isomorphism
7336:to the linear map
7281:universal property
7235:tensor contraction
7186:, then a trace on
7155:
6995:
6955:
6911:
6833:
6745:
6714:
6685:
6635:
6615:
6591:
6547:
6477:
6447:
6350:
6279:
6092:
6032:
5982:times scalars) so
5972:
5940:
5895:
5823:
5729:
5685:" to obtain a map
5671:
5625:
5565:
5482:
5457:simple Lie algebra
5423:
5320:
5286:
5252:
5065:
5011:
4972:
4912:
4867:
4832:
4674:of linear maps on
4666:between the space
4593:
4472:divergence theorem
4395:
4325:matrix exponential
4300:
4200:
4094:
4080:
3986:
3975:
3870:
3837:
3817:
3781:
3719:
3622:
3620:
3436:permutation matrix
3409:
3203:
3108:
3088:
3051:
3032:are proportional.
3022:
3002:
2982:
2925:
2901:
2899:
2685:
2591:
2424:
2337:
2162:
2085:
2048:
2004:
1809:
1662:
1350:
1089:Trace of a product
1073:
979:
977:
818:
673:
667:
598:
463:be a matrix, with
389:
10018:978-0-521-54823-6
9210:
9146:
8212:given by sending
8119:given by sending
7961:
7951:
7886:
7876:
7671:
7661:
7604:
7579:
7328:given by sending
7263:given by sending
7233:The operation of
6946:
6880:orthonormal basis
6857:compact operators
6791:
6734:
6706:
6671:
6601:of a Lie algebra
6207:
5877:
5812:
5681:of "inclusion of
5598:
5385:
5323:{\displaystyle K}
4842:, and any random
4801:Given any matrix
4769:corresponding to
4733:is defined to be
4411:. Given a matrix
4071:
4015:triangular matrix
3966:
3840:{\displaystyle A}
3658:, equals its own
3646:idempotent matrix
3464:projection matrix
3438:is the number of
3352:The trace of the
3111:{\displaystyle f}
3005:{\displaystyle f}
2937:linear functional
2928:{\displaystyle f}
2611:Kronecker product
2609:The trace of the
2153:
2131:
1715:complex conjugate
1658:
1011:A matrix and its
428:. The entries of
130:
129:
122:
104:
18:Trace of a matrix
16:(Redirected from
10115:
10084:
10058:
10045:Cengage Learning
10043:(4th ed.).
10030:
9990:
9973:. Translated by
9967:Gantmacher, F.R.
9958:
9957:
9907:
9894:
9893:
9875:
9869:
9868:
9828:
9822:
9821:
9804:(3): 1059–1076.
9789:
9783:
9782:
9764:
9758:
9757:
9739:
9726:
9725:
9723:
9722:
9672:"Trace (matrix)"
9664:
9651:
9650:
9648:
9647:
9633:
9608:
9606:
9594:
9581:
9575:
9565:
9563:
9562:
9557:
9555:
9554:
9549:
9548:
9530:
9524:
9521:
9519:
9518:
9513:
9508:
9507:
9502:
9501:
9484:
9482:
9481:
9476:
9434:
9432:
9431:
9426:
9378:
9376:
9375:
9370:
9359:
9358:
9353:
9352:
9339:
9338:
9333:
9332:
9315:
9313:
9312:
9307:
9299:
9285:
9281:
9280:
9261:
9257:
9256:
9240:
9239:
9227:
9218:
9206:
9202:
9201:
9182:
9181:
9169:
9160:
9139:
9122:
9120:
9119:
9114:
9112:
9108:
9107:
9088:
9084:
9083:
9057:
9055:
9054:
9049:
9031:
9029:
9028:
9023:
9015:
9011:
9010:
8984:
8982:
8981:
8976:
8974:
8973:
8953:
8947:
8945:
8929:
8927:
8926:
8921:
8916:
8915:
8877:
8862:
8860:
8859:
8854:
8849:
8848:
8810:
8801:
8800:
8762:
8751:For example, if
8749:
8743:
8742:
8740:
8739:
8734:
8726:
8721:
8704:
8703:
8695:
8691:
8690:
8685:
8673:
8668:
8650:
8649:
8637:
8636:
8623:
8618:
8602:
8597:
8579:
8578:
8566:
8565:
8552:
8547:
8531:
8526:
8508:
8507:
8499:
8495:
8494:
8489:
8477:
8472:
8451:
8446:
8420:
8384:Specht's theorem
8336:
8331:
8320:, and the trace
8318:coevaluation map
8315:
8303:
8299:
8298:
8282:
8275:
8249:
8233:
8223:
8211:
8195:
8175:
8156:
8138:
8118:
8091:
8081:
8070:
8058:
8046:
8042:
8035:
8033:
8032:
8027:
8019:
8018:
8006:
8005:
7993:
7992:
7980:
7979:
7969:
7959:
7944:
7943:
7931:
7930:
7918:
7917:
7905:
7904:
7894:
7884:
7841:
7802:
7756:
7752:
7745:
7743:
7742:
7737:
7735:
7734:
7722:
7721:
7709:
7708:
7690:
7689:
7679:
7669:
7657:
7656:
7647:
7643:
7642:
7641:
7623:
7622:
7612:
7598:
7597:
7587:
7541:
7537:
7533:
7510:
7501:
7478:
7469:
7458:
7447:
7436:
7425:
7421:
7417:
7402:
7390:
7380:
7372:
7368:
7349:
7335:
7327:
7305:
7295:
7278:
7270:
7262:
7248:
7218:
7204:
7200:
7189:
7185:
7177:
7164:
7162:
7161:
7156:
7151:
7147:
7134:
7133:
7116:
7115:
7103:
7099:
7086:
7085:
7068:
7067:
7033:
7029:
7025:
7015:
7004:
7002:
7001:
6996:
6991:
6987:
6986:
6985:
6970:
6969:
6954:
6920:
6918:
6917:
6912:
6910:
6909:
6900:
6899:
6877:
6842:
6840:
6839:
6834:
6826:
6812:
6798:
6797:
6792:
6789:
6780:
6772:
6754:
6752:
6751:
6746:
6735:
6732:
6723:
6721:
6720:
6715:
6713:
6712:
6707:
6704:
6694:
6692:
6691:
6686:
6672:
6669:
6664:
6663:
6644:
6642:
6641:
6636:
6624:
6622:
6621:
6616:
6614:
6613:
6600:
6598:
6597:
6592:
6581:
6580:
6556:
6554:
6553:
6548:
6537:
6532:
6510:trace orthogonal
6507:
6501:
6492:
6486:
6484:
6483:
6478:
6476:
6475:
6456:
6454:
6453:
6448:
6440:
6432:
6424:
6398:
6390:
6382:
6359:
6357:
6356:
6351:
6343:
6338:
6318:
6310:
6288:
6286:
6285:
6280:
6278:
6273:
6265:
6260:
6249:
6241:
6230:
6222:
6208:
6205:
6196:
6179:
6150:
6142:
6124:
6118:
6101:
6099:
6098:
6093:
6088:
6087:
6075:
6074:
6062:
6061:
6045:
6041:
6039:
6038:
6033:
6025:
6024:
6019:
6018:
6005:
6004:
5999:
5998:
5981:
5979:
5978:
5973:
5968:
5949:
5947:
5946:
5941:
5921:
5920:
5904:
5902:
5901:
5896:
5888:
5887:
5878:
5870:
5868:
5867:
5855:
5854:
5832:
5830:
5829:
5824:
5813:
5805:
5803:
5802:
5797:
5796:
5783:
5782:
5777:
5776:
5746:
5742:
5738:
5736:
5735:
5730:
5728:
5727:
5722:
5721:
5708:
5707:
5702:
5701:
5680:
5678:
5677:
5672:
5670:
5669:
5664:
5663:
5634:
5632:
5631:
5626:
5621:
5613:
5599:
5591:
5586:
5574:
5572:
5571:
5566:
5558:
5557:
5552:
5551:
5538:
5537:
5532:
5531:
5491:
5489:
5488:
5483:
5481:
5480:
5475:
5474:
5453:
5452:
5445:
5444:
5432:
5430:
5429:
5424:
5419:
5418:
5413:
5412:
5399:
5391:
5386:
5383:
5369:
5361:
5337:
5333:
5329:
5327:
5326:
5321:
5309:
5299:
5295:
5293:
5292:
5287:
5285:
5284:
5279:
5278:
5261:
5259:
5258:
5253:
5245:
5244:
5239:
5238:
5200:
5190:
5170:
5166:
5162:
5116:
5074:
5072:
5071:
5066:
5064:
5063:
5054:
5053:
5049:
5020:
5018:
5017:
5012:
4981:
4979:
4978:
4973:
4944:
4943:
4921:
4919:
4918:
4913:
4902:
4901:
4876:
4874:
4873:
4868:
4866:
4865:
4860:
4841:
4839:
4838:
4833:
4831:
4830:
4819:
4783:
4776:
4768:
4757:
4743:
4732:
4722:
4715:
4711:
4707:
4703:
4695:
4688:
4677:
4673:
4657:similar matrices
4654:
4650:
4642:
4627:
4623:
4602:
4600:
4599:
4594:
4586:
4563:
4536:
4528:
4520:
4508:
4500:
4494:
4490:
4484:
4466:
4458:
4448:
4442:
4428:
4422:
4416:
4404:
4402:
4401:
4396:
4385:
4353:
4309:
4307:
4306:
4301:
4299:
4298:
4292:
4278:
4264:
4263:
4245:
4222:Jacobi's formula
4209:
4207:
4206:
4201:
4193:
4164:
4153:
4129:
4123:
4103:
4101:
4100:
4095:
4090:
4089:
4079:
4064:
4042:
4032:
4004:
3995:
3993:
3992:
3987:
3985:
3984:
3974:
3959:
3931:
3921:
3897:
3879:
3877:
3876:
3871:
3869:
3868:
3846:
3844:
3843:
3838:
3826:
3824:
3823:
3818:
3801:The trace of an
3790:
3788:
3787:
3782:
3768:
3764:
3763:
3758:
3739:
3734:
3729:
3724:
3701:
3697:
3688:
3682:
3678:
3667:nilpotent matrix
3657:
3648:, i.e. one with
3640:
3631:
3629:
3628:
3623:
3621:
3611:
3590:
3586:
3585:
3584:
3578:
3555:
3554:
3553:
3547:
3541:
3540:
3532:
3528:
3527:
3522:
3521:
3520:
3514:
3502:
3490:
3489:
3488:
3482:
3458:
3452:
3429:Hermitian matrix
3418:
3416:
3415:
3410:
3402:
3398:
3397:
3392:
3368:
3361:
3330:
3314:
3310:
3298:
3294:
3288:
3278:
3268:
3262:
3244:
3238:
3228:
3212:
3210:
3209:
3204:
3202:
3201:
3191:
3186:
3165:
3141:
3135:
3117:
3115:
3114:
3109:
3097:
3095:
3094:
3089:
3078:
3060:
3058:
3057:
3052:
3031:
3029:
3028:
3023:
3011:
3009:
3008:
3003:
2991:
2989:
2988:
2983:
2934:
2932:
2931:
2926:
2910:
2908:
2907:
2902:
2900:
2890:
2885:
2861:
2856:
2832:
2805:
2778:
2758:
2734:
2726:
2694:
2692:
2691:
2686:
2678:
2661:
2641:
2633:
2600:
2598:
2597:
2592:
2584:
2579:
2574:
2554:
2549:
2544:
2527:
2523:
2522:
2521:
2515:
2511:
2510:
2505:
2500:
2473:
2468:
2463:
2433:
2431:
2430:
2425:
2417:
2412:
2407:
2387:
2382:
2377:
2346:
2344:
2343:
2338:
2330:
2325:
2320:
2315:
2295:
2290:
2285:
2280:
2260:
2255:
2250:
2245:
2225:
2220:
2215:
2210:
2182:invariant under
2171:
2169:
2168:
2163:
2161:
2156:
2155:
2154:
2148:
2139:
2135:
2134:
2133:
2132:
2126:
2120:
2094:
2092:
2091:
2086:
2084:
2083:
2078:
2069:
2057:
2055:
2054:
2049:
2047:
2046:
2041:
2032:
2013:
2011:
2010:
2005:
1997:
1980:
1976:
1975:
1974:
1966:
1957:
1952:
1930:
1926:
1922:
1917:
1909:
1908:
1900:
1876:
1870:
1864:
1845:
1833:
1827:
1818:
1816:
1815:
1810:
1805:
1800:
1780:
1775:
1751:
1741:
1731:
1725:
1712:
1683:
1677:
1671:
1669:
1668:
1663:
1656:
1655:
1654:
1649:
1645:
1641:
1621:
1620:
1615:
1611:
1607:
1584:
1580:
1579:
1574:
1558:
1554:
1553:
1548:
1529:
1528:
1523:
1519:
1515:
1510:
1461:
1455:
1445:
1430:
1412:
1406:
1391:
1385:
1375:
1371:
1359:
1357:
1356:
1351:
1345:
1344:
1332:
1331:
1318:
1313:
1297:
1292:
1274:
1270:
1269:
1268:
1267:
1261:
1255:
1236:
1232:
1231:
1226:
1225:
1224:
1218:
1198:
1194:
1193:
1192:
1191:
1185:
1179:
1160:
1156:
1155:
1150:
1149:
1148:
1142:
1119:matrices, then:
1118:
1108:
1102:
1095:Hadamard product
1082:
1080:
1079:
1074:
1069:
1065:
1064:
1063:
1057:
1035:
1007:
1000:
994:
988:
986:
985:
980:
978:
971:
944:
920:
900:
876:
868:
836:Basic properties
827:
825:
824:
819:
784:
783:
771:
770:
758:
757:
745:
744:
731:
726:
705:
682:
680:
679:
674:
672:
671:
603:
602:
595:
594:
583:
582:
571:
570:
557:
556:
545:
544:
533:
532:
519:
518:
507:
506:
495:
494:
474:
462:
448:
433:
427:
421:
419:
414:
412:
407:
398:
396:
395:
390:
388:
387:
366:
365:
353:
352:
340:
339:
326:
321:
300:
279:
271:
248:Jacobi's formula
229:similar matrices
226:
220:
214:
187:
176:
166:
156:
148:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
10123:
10122:
10118:
10117:
10116:
10114:
10113:
10112:
10088:
10087:
10069:
10066:
10061:
10055:
10054:978-003010567-8
10035:
10019:
10005:Matrix Analysis
9995:
9965:
9961:
9938:
9920:Springer-Verlag
9909:
9908:
9897:
9890:
9877:
9876:
9872:
9843:(2): 8:1–8:34.
9830:
9829:
9825:
9791:
9790:
9786:
9779:
9769:Matrix Analysis
9766:
9765:
9761:
9754:
9741:
9740:
9729:
9720:
9718:
9700:
9666:
9665:
9654:
9645:
9643:
9635:
9634:
9621:
9617:
9612:
9611:
9598:
9584:
9582:
9578:
9572:derived algebra
9539:
9534:
9533:
9531:
9527:
9492:
9487:
9486:
9437:
9436:
9435:(equivalently,
9381:
9380:
9343:
9323:
9318:
9317:
9272:
9268:
9248:
9244:
9228:
9190:
9186:
9170:
9124:
9123:
9099:
9095:
9072:
9068:
9060:
9059:
9034:
9033:
9032:if and only if
8999:
8995:
8987:
8986:
8962:
8957:
8956:
8954:
8950:
8931:
8910:
8909:
8904:
8898:
8897:
8892:
8882:
8865:
8864:
8843:
8842:
8837:
8831:
8830:
8825:
8815:
8795:
8794:
8789:
8783:
8782:
8777:
8767:
8753:
8752:
8750:
8746:
8680:
8676:
8675:
8638:
8625:
8567:
8554:
8484:
8480:
8479:
8428:
8427:
8421:
8417:
8412:
8355:
8347:category theory
8329:
8321:
8314:
8305:
8304:the linear map
8301:
8296:
8284:
8280:
8251:
8235:
8225:
8213:
8197:
8177:
8158:
8140:
8120:
8094:
8083:
8075:
8060:
8048:
8044:
8040:
8010:
7997:
7984:
7971:
7935:
7922:
7909:
7896:
7847:
7846:
7839:
7830:
7821:
7812:
7804:
7801:
7792:
7783:
7770:
7758:
7754:
7750:
7726:
7713:
7700:
7681:
7648:
7633:
7614:
7603:
7599:
7589:
7547:
7546:
7539:
7535:
7532:
7519:
7508:
7503:
7500:
7487:
7476:
7471:
7468:
7460:
7457:
7449:
7446:
7438:
7435:
7427:
7423:
7419:
7407:
7392:
7382:
7378:
7370:
7351:
7337:
7329:
7310:
7297:
7287:
7272:
7264:
7250:
7246:
7243:
7206:
7202:
7191:
7187:
7183:
7175:
7125:
7124:
7120:
7107:
7077:
7076:
7072:
7059:
7036:
7035:
7031:
7027:
7017:
7013:
6977:
6961:
6960:
6956:
6923:
6922:
6901:
6891:
6883:
6882:
6875:
6869:Hilbert–Schmidt
6849:
6847:Generalizations
6787:
6757:
6756:
6726:
6725:
6702:
6697:
6696:
6695:The trace form
6647:
6646:
6627:
6626:
6603:
6602:
6561:
6560:
6514:
6513:
6508:are said to be
6503:
6497:
6491:
6462:
6461:
6460:
6364:
6363:
6298:
6297:
6127:
6126:
6120:
6114:
6107:
6079:
6066:
6053:
6048:
6047:
6043:
6009:
5989:
5984:
5983:
5956:
5955:
5912:
5907:
5906:
5879:
5859:
5846:
5835:
5834:
5787:
5767:
5756:
5755:
5744:
5740:
5712:
5692:
5687:
5686:
5654:
5643:
5642:
5577:
5576:
5542:
5522:
5517:
5516:
5492:, which is the
5465:
5460:
5459:
5450:
5449:
5442:
5441:
5403:
5340:
5339:
5335:
5334:of scalars; as
5331:
5312:
5311:
5301:
5297:
5269:
5264:
5263:
5229:
5218:
5217:
5214:
5206:quadratic forms
5192:
5172:
5168:
5164:
5141:
5114:
5092:diagonal matrix
5088:
5055:
5034:
5023:
5022:
4988:
4987:
4935:
4924:
4923:
4893:
4879:
4878:
4855:
4844:
4843:
4814:
4803:
4802:
4795:
4790:
4778:
4774:
4759:
4745:
4734:
4724:
4717:
4713:
4709:
4705:
4701:
4690:
4679:
4675:
4667:
4652:
4651:and describing
4648:
4640:
4625:
4611:
4608:
4542:
4541:
4534:
4522:
4510:
4506:
4496:
4495:is a region in
4492:
4486:
4475:
4460:
4453:
4444:
4430:
4424:
4418:
4412:
4329:
4328:
4320:of the matrix.
4227:
4226:
4138:
4137:
4132:identity matrix
4125:
4119:
4116:
4081:
4049:
4048:
4038:
4030:
4024:
4018:
4000:
3997:
3976:
3941:
3940:
3927:
3919:
3913:
3907:
3906:entries and if
3893:
3890:
3854:
3849:
3848:
3829:
3828:
3803:
3802:
3799:
3794:
3753:
3749:
3715:
3704:
3703:
3699:
3692:
3684:
3680:
3672:
3665:The trace of a
3649:
3638:
3633:
3619:
3618:
3591:
3573:
3569:
3557:
3556:
3542:
3509:
3508:
3504:
3503:
3491:
3477:
3468:
3467:
3462:The trace of a
3454:
3451:
3443:
3434:The trace of a
3427:The trace of a
3387:
3383:
3372:
3371:
3366:
3363:identity matrix
3353:
3348:
3325:
3319:
3312:
3300:
3296:
3290:
3284:
3270:
3264:
3258:
3255:
3240:
3234:
3227:
3221:
3217:
3214:
3193:
3147:
3146:
3137:
3127:
3124:
3100:
3099:
3063:
3062:
3037:
3036:
3014:
3013:
2994:
2993:
2941:
2940:
2917:
2916:
2898:
2897:
2865:
2840:
2839:
2809:
2786:
2785:
2738:
2704:
2703:
2700:
2615:
2614:
2607:
2495:
2491:
2490:
2486:
2445:
2444:
2359:
2358:
2352:cyclic property
2348:
2192:
2191:
2184:circular shifts
2178:
2176:Cyclic property
2173:
2143:
2121:
2115:
2111:
2100:
2099:
2073:
2060:
2059:
2036:
2023:
2022:
1961:
1944:
1940:
1895:
1894:
1890:
1879:
1878:
1872:
1866:
1851:
1835:
1829:
1823:
1820:
1757:
1756:
1743:
1733:
1727:
1721:
1708:
1690:matrix calculus
1679:
1673:
1627:
1623:
1622:
1593:
1589:
1588:
1569:
1565:
1543:
1539:
1496:
1492:
1491:
1480:
1479:
1457:
1451:
1436:
1414:
1408:
1397:
1387:
1381:
1373:
1363:
1333:
1320:
1256:
1250:
1246:
1213:
1212:
1208:
1180:
1174:
1170:
1137:
1136:
1132:
1121:
1120:
1110:
1104:
1098:
1091:
1052:
1048:
1017:
1016:
1005:
996:
990:
976:
975:
948:
925:
924:
880:
846:
845:
840:The trace is a
838:
833:
775:
762:
749:
733:
687:
686:
666:
665:
657:
652:
646:
645:
640:
635:
629:
628:
623:
618:
608:
597:
596:
586:
584:
574:
572:
562:
559:
558:
548:
546:
536:
534:
524:
521:
520:
510:
508:
498:
496:
486:
479:
465:
464:
458:
455:
446:
440:complex numbers
429:
423:
417:
416:
410:
409:
405:
400:
376:
357:
344:
328:
282:
281:
275:
263:
256:
233:linear operator
222:
216:
204:
181:
168:
162:
150:
144:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
10121:
10119:
10111:
10110:
10105:
10100:
10098:Linear algebra
10090:
10089:
10086:
10085:
10065:
10064:External links
10062:
10060:
10059:
10053:
10032:
10031:
10017:
9992:
9991:
9977:New York, NY:
9962:
9960:
9959:
9936:
9912:Quantum groups
9895:
9889:978-1470417048
9888:
9870:
9823:
9784:
9777:
9759:
9752:
9727:
9698:
9652:
9618:
9616:
9613:
9610:
9609:
9596:if and only if
9576:
9553:
9547:
9544:
9525:
9511:
9506:
9500:
9497:
9474:
9471:
9468:
9465:
9462:
9459:
9456:
9453:
9450:
9447:
9444:
9424:
9421:
9418:
9415:
9412:
9409:
9406:
9403:
9400:
9397:
9394:
9391:
9388:
9368:
9365:
9362:
9357:
9351:
9348:
9342:
9337:
9331:
9328:
9305:
9302:
9298:
9294:
9291:
9288:
9284:
9279:
9275:
9271:
9267:
9264:
9260:
9255:
9251:
9247:
9243:
9238:
9235:
9231:
9226:
9222:
9217:
9213:
9209:
9205:
9200:
9197:
9193:
9189:
9185:
9180:
9177:
9173:
9168:
9164:
9159:
9156:
9153:
9149:
9145:
9142:
9138:
9134:
9131:
9111:
9106:
9102:
9098:
9094:
9091:
9087:
9082:
9079:
9075:
9071:
9067:
9047:
9044:
9041:
9021:
9018:
9014:
9009:
9006:
9002:
8998:
8994:
8972:
8969:
8965:
8948:
8919:
8914:
8908:
8905:
8903:
8900:
8899:
8896:
8893:
8891:
8888:
8887:
8885:
8880:
8876:
8873:
8852:
8847:
8841:
8838:
8836:
8833:
8832:
8829:
8826:
8824:
8821:
8820:
8818:
8813:
8809:
8804:
8799:
8793:
8790:
8788:
8785:
8784:
8781:
8778:
8776:
8773:
8772:
8770:
8765:
8761:
8744:
8732:
8729:
8725:
8720:
8716:
8713:
8710:
8707:
8702:
8699:
8694:
8689:
8684:
8679:
8672:
8667:
8664:
8661:
8657:
8653:
8648:
8645:
8641:
8635:
8632:
8628:
8622:
8617:
8614:
8611:
8607:
8601:
8596:
8593:
8590:
8586:
8582:
8577:
8574:
8570:
8564:
8561:
8557:
8551:
8546:
8543:
8540:
8536:
8530:
8525:
8522:
8519:
8515:
8511:
8506:
8503:
8498:
8493:
8488:
8483:
8476:
8471:
8468:
8465:
8461:
8457:
8454:
8450:
8445:
8441:
8438:
8435:
8424:matrix product
8414:
8413:
8411:
8408:
8407:
8406:
8401:
8396:
8394:Trace identity
8391:
8386:
8381:
8379:Singular trace
8376:
8371:
8366:
8361:
8354:
8351:
8339:evaluation map
8310:
8037:
8036:
8025:
8022:
8017:
8013:
8009:
8004:
8000:
7996:
7991:
7987:
7983:
7978:
7974:
7968:
7964:
7958:
7954:
7950:
7947:
7942:
7938:
7934:
7929:
7925:
7921:
7916:
7912:
7908:
7903:
7899:
7893:
7889:
7883:
7879:
7875:
7872:
7869:
7866:
7863:
7860:
7857:
7854:
7835:
7826:
7817:
7808:
7797:
7788:
7779:
7766:
7747:
7746:
7733:
7729:
7725:
7720:
7716:
7712:
7707:
7703:
7699:
7696:
7693:
7688:
7684:
7678:
7674:
7668:
7664:
7660:
7655:
7651:
7646:
7640:
7636:
7632:
7629:
7626:
7621:
7617:
7611:
7607:
7602:
7596:
7592:
7586:
7582:
7578:
7575:
7572:
7569:
7566:
7563:
7560:
7557:
7554:
7528:
7515:
7496:
7483:
7464:
7453:
7442:
7431:
7285:tensor product
7271:to the scalar
7242:
7239:
7154:
7150:
7146:
7143:
7140:
7137:
7132:
7128:
7123:
7119:
7114:
7110:
7106:
7102:
7098:
7095:
7092:
7089:
7084:
7080:
7075:
7071:
7066:
7062:
7058:
7055:
7052:
7049:
7046:
7043:
6994:
6990:
6984:
6980:
6976:
6973:
6968:
6964:
6959:
6953:
6949:
6945:
6942:
6939:
6936:
6933:
6930:
6908:
6904:
6898:
6894:
6890:
6867:is called the
6865:Frobenius norm
6861:Hilbert spaces
6848:
6845:
6832:
6829:
6825:
6821:
6818:
6815:
6811:
6807:
6804:
6801:
6796:
6786:
6783:
6779:
6775:
6771:
6767:
6764:
6744:
6741:
6738:
6711:
6684:
6681:
6678:
6675:
6667:
6662:
6657:
6654:
6634:
6612:
6590:
6587:
6584:
6579:
6574:
6571:
6568:
6546:
6543:
6540:
6536:
6531:
6527:
6524:
6521:
6487:
6474:
6471:
6446:
6443:
6439:
6435:
6431:
6427:
6423:
6419:
6416:
6413:
6410:
6407:
6404:
6401:
6397:
6393:
6389:
6385:
6381:
6377:
6374:
6371:
6349:
6346:
6342:
6337:
6333:
6330:
6327:
6324:
6321:
6317:
6313:
6309:
6305:
6289:is called the
6277:
6272:
6268:
6264:
6259:
6255:
6252:
6248:
6244:
6240:
6236:
6233:
6229:
6225:
6221:
6217:
6214:
6211:
6202:
6199:
6195:
6191:
6188:
6185:
6182:
6178:
6174:
6171:
6168:
6165:
6162:
6159:
6156:
6153:
6149:
6145:
6141:
6137:
6134:
6106:
6105:Bilinear forms
6103:
6091:
6086:
6082:
6078:
6073:
6069:
6065:
6060:
6056:
6031:
6028:
6023:
6017:
6014:
6008:
6003:
5997:
5994:
5971:
5967:
5963:
5939:
5936:
5933:
5930:
5927:
5924:
5919:
5915:
5894:
5891:
5886:
5882:
5876:
5873:
5866:
5862:
5858:
5853:
5849:
5845:
5842:
5822:
5819:
5816:
5811:
5808:
5801:
5795:
5792:
5786:
5781:
5775:
5772:
5766:
5763:
5743:. Dividing by
5726:
5720:
5717:
5711:
5706:
5700:
5697:
5668:
5662:
5659:
5653:
5650:
5624:
5620:
5616:
5612:
5608:
5605:
5602:
5597:
5594:
5589:
5585:
5564:
5561:
5556:
5550:
5547:
5541:
5536:
5530:
5527:
5515:decomposition
5479:
5473:
5470:
5422:
5417:
5411:
5408:
5402:
5398:
5394:
5390:
5381:
5378:
5375:
5372:
5368:
5364:
5360:
5356:
5353:
5350:
5347:
5319:
5283:
5277:
5274:
5251:
5248:
5243:
5237:
5234:
5228:
5225:
5213:
5210:
5099:complex matrix
5087:
5084:
5062:
5058:
5052:
5048:
5044:
5041:
5037:
5033:
5030:
5010:
5007:
5004:
5001:
4998:
4995:
4971:
4968:
4965:
4962:
4959:
4956:
4953:
4950:
4947:
4942:
4938:
4934:
4931:
4911:
4908:
4905:
4900:
4896:
4892:
4889:
4886:
4864:
4859:
4854:
4851:
4829:
4826:
4823:
4818:
4813:
4810:
4794:
4791:
4789:
4786:
4607:
4604:
4592:
4589:
4585:
4581:
4578:
4575:
4572:
4569:
4566:
4562:
4558:
4555:
4552:
4549:
4394:
4391:
4388:
4384:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4352:
4348:
4345:
4342:
4339:
4336:
4297:
4291:
4287:
4284:
4281:
4277:
4273:
4270:
4267:
4262:
4257:
4254:
4251:
4248:
4244:
4240:
4237:
4234:
4199:
4196:
4192:
4189:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4163:
4160:
4156:
4152:
4148:
4145:
4115:
4112:
4093:
4088:
4084:
4078:
4074:
4070:
4067:
4063:
4059:
4056:
4028:
4022:
3983:
3979:
3973:
3969:
3965:
3962:
3958:
3954:
3951:
3948:
3938:
3917:
3911:
3889:
3886:
3867:
3864:
3861:
3857:
3836:
3816:
3813:
3810:
3798:
3795:
3793:
3792:
3780:
3777:
3774:
3771:
3767:
3762:
3757:
3752:
3748:
3745:
3742:
3738:
3733:
3728:
3723:
3718:
3714:
3711:
3689:is nilpotent.
3663:
3642:
3641:is idempotent.
3636:
3617:
3614:
3610:
3606:
3603:
3600:
3597:
3594:
3592:
3589:
3583:
3577:
3572:
3568:
3565:
3562:
3559:
3558:
3552:
3546:
3539:
3536:
3531:
3526:
3519:
3513:
3507:
3501:
3497:
3494:
3492:
3487:
3481:
3476:
3475:
3460:
3447:
3432:
3425:
3420:This leads to
3408:
3405:
3401:
3396:
3391:
3386:
3382:
3379:
3349:
3347:
3344:
3321:
3254:
3251:
3223:
3219:
3200:
3196:
3190:
3185:
3182:
3179:
3175:
3171:
3168:
3164:
3160:
3157:
3154:
3144:
3123:
3120:
3107:
3087:
3084:
3081:
3077:
3073:
3070:
3050:
3047:
3044:
3021:
3001:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2924:
2896:
2893:
2889:
2884:
2880:
2877:
2874:
2871:
2868:
2866:
2864:
2860:
2855:
2851:
2848:
2845:
2842:
2841:
2838:
2835:
2831:
2827:
2824:
2821:
2818:
2815:
2812:
2810:
2808:
2804:
2800:
2797:
2794:
2791:
2788:
2787:
2784:
2781:
2777:
2773:
2770:
2767:
2764:
2761:
2757:
2753:
2750:
2747:
2744:
2741:
2739:
2737:
2733:
2729:
2725:
2721:
2718:
2715:
2712:
2711:
2699:
2696:
2684:
2681:
2677:
2673:
2670:
2667:
2664:
2660:
2656:
2653:
2650:
2647:
2644:
2640:
2636:
2632:
2628:
2625:
2622:
2606:
2603:
2590:
2587:
2583:
2578:
2573:
2569:
2566:
2563:
2560:
2557:
2553:
2548:
2543:
2539:
2536:
2533:
2530:
2526:
2520:
2514:
2509:
2504:
2499:
2494:
2489:
2485:
2482:
2479:
2476:
2472:
2467:
2462:
2458:
2455:
2452:
2423:
2420:
2416:
2411:
2406:
2402:
2399:
2396:
2393:
2390:
2386:
2381:
2376:
2372:
2369:
2366:
2336:
2333:
2329:
2324:
2319:
2314:
2310:
2307:
2304:
2301:
2298:
2294:
2289:
2284:
2279:
2275:
2272:
2269:
2266:
2263:
2259:
2254:
2249:
2244:
2240:
2237:
2234:
2231:
2228:
2224:
2219:
2214:
2209:
2205:
2202:
2199:
2189:
2177:
2174:
2160:
2147:
2142:
2138:
2125:
2119:
2114:
2110:
2107:
2097:
2082:
2077:
2072:
2068:
2045:
2040:
2035:
2031:
2003:
2000:
1996:
1992:
1989:
1986:
1983:
1979:
1973:
1970:
1965:
1960:
1956:
1951:
1947:
1943:
1939:
1936:
1933:
1929:
1925:
1921:
1916:
1912:
1907:
1904:
1899:
1893:
1889:
1886:
1808:
1804:
1799:
1795:
1792:
1789:
1786:
1783:
1779:
1774:
1770:
1767:
1764:
1754:
1661:
1653:
1648:
1644:
1640:
1636:
1633:
1630:
1626:
1619:
1614:
1610:
1606:
1602:
1599:
1596:
1592:
1587:
1583:
1578:
1573:
1568:
1564:
1561:
1557:
1552:
1547:
1542:
1538:
1535:
1532:
1527:
1522:
1518:
1514:
1509:
1505:
1502:
1499:
1495:
1490:
1487:
1472:Frobenius norm
1349:
1343:
1340:
1336:
1330:
1327:
1323:
1317:
1312:
1309:
1306:
1302:
1296:
1291:
1288:
1285:
1281:
1277:
1273:
1266:
1260:
1254:
1249:
1245:
1242:
1239:
1235:
1230:
1223:
1217:
1211:
1207:
1204:
1201:
1197:
1190:
1184:
1178:
1173:
1169:
1166:
1163:
1159:
1154:
1147:
1141:
1135:
1131:
1128:
1090:
1087:
1072:
1068:
1062:
1056:
1051:
1047:
1044:
1041:
1038:
1034:
1030:
1027:
1024:
974:
970:
966:
963:
960:
957:
954:
951:
949:
947:
943:
939:
936:
933:
930:
927:
926:
923:
919:
915:
912:
909:
906:
903:
899:
895:
892:
889:
886:
883:
881:
879:
875:
871:
867:
863:
860:
857:
854:
853:
842:linear mapping
837:
834:
832:
829:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
782:
778:
774:
769:
765:
761:
756:
752:
748:
743:
740:
736:
730:
725:
722:
719:
715:
711:
708:
704:
700:
697:
694:
670:
664:
661:
658:
656:
653:
651:
648:
647:
644:
641:
639:
636:
634:
631:
630:
627:
624:
622:
619:
617:
614:
613:
611:
606:
601:
593:
589:
585:
581:
577:
573:
569:
565:
561:
560:
555:
551:
547:
543:
539:
535:
531:
527:
523:
522:
517:
513:
509:
505:
501:
497:
493:
489:
485:
484:
482:
477:
473:
454:
451:
403:
386:
383:
379:
375:
372:
369:
364:
360:
356:
351:
347:
343:
338:
335:
331:
325:
320:
317:
314:
310:
306:
303:
299:
295:
292:
289:
280:is defined as
255:
252:
194:Pauli Matrices
134:linear algebra
128:
127:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
10120:
10109:
10106:
10104:
10103:Matrix theory
10101:
10099:
10096:
10095:
10093:
10082:
10078:
10077:
10072:
10068:
10067:
10063:
10056:
10050:
10046:
10042:
10038:
10034:
10033:
10028:
10024:
10020:
10014:
10010:
10006:
10002:
10001:Johnson, C.R.
9998:
9994:
9993:
9988:
9984:
9980:
9976:
9972:
9968:
9964:
9963:
9955:
9951:
9947:
9943:
9939:
9937:0-387-94370-6
9933:
9929:
9925:
9921:
9917:
9913:
9906:
9904:
9902:
9900:
9896:
9891:
9885:
9881:
9874:
9871:
9866:
9862:
9858:
9854:
9850:
9846:
9842:
9838:
9834:
9827:
9824:
9819:
9815:
9811:
9807:
9803:
9799:
9795:
9788:
9785:
9780:
9778:9780521839402
9774:
9770:
9763:
9760:
9755:
9753:9780070605022
9749:
9745:
9738:
9736:
9734:
9732:
9728:
9717:
9713:
9709:
9705:
9701:
9699:1-58488-347-2
9695:
9691:
9687:
9683:
9679:
9678:
9673:
9669:
9663:
9661:
9659:
9657:
9653:
9642:
9638:
9632:
9630:
9628:
9626:
9624:
9620:
9614:
9605:
9601:
9597:
9592:
9588:
9580:
9577:
9573:
9569:
9551:
9529:
9526:
9509:
9504:
9472:
9469:
9460:
9457:
9454:
9445:
9442:
9419:
9416:
9410:
9407:
9404:
9398:
9395:
9389:
9386:
9366:
9363:
9360:
9355:
9340:
9335:
9303:
9289:
9286:
9282:
9277:
9273:
9269:
9265:
9262:
9258:
9253:
9249:
9245:
9241:
9236:
9233:
9215:
9211:
9207:
9203:
9198:
9195:
9191:
9187:
9183:
9178:
9175:
9157:
9154:
9151:
9147:
9143:
9129:
9109:
9104:
9100:
9096:
9092:
9089:
9085:
9080:
9077:
9073:
9069:
9065:
9045:
9042:
9039:
9019:
9016:
9012:
9007:
9004:
9000:
8996:
8992:
8970:
8967:
8963:
8952:
8949:
8943:
8939:
8935:
8917:
8912:
8906:
8901:
8894:
8889:
8883:
8878:
8850:
8845:
8839:
8834:
8827:
8822:
8816:
8811:
8802:
8797:
8791:
8786:
8779:
8774:
8768:
8763:
8748:
8745:
8730:
8711:
8708:
8705:
8700:
8697:
8692:
8677:
8670:
8665:
8662:
8659:
8655:
8651:
8646:
8643:
8639:
8633:
8630:
8626:
8620:
8615:
8612:
8609:
8605:
8599:
8594:
8591:
8588:
8584:
8580:
8575:
8572:
8568:
8562:
8559:
8555:
8549:
8544:
8541:
8538:
8534:
8528:
8523:
8520:
8517:
8513:
8509:
8504:
8501:
8496:
8481:
8474:
8469:
8466:
8463:
8459:
8455:
8436:
8433:
8425:
8419:
8416:
8409:
8405:
8402:
8400:
8397:
8395:
8392:
8390:
8387:
8385:
8382:
8380:
8377:
8375:
8372:
8370:
8367:
8365:
8362:
8360:
8357:
8356:
8352:
8350:
8348:
8344:
8340:
8335:
8328:
8324:
8319:
8313:
8308:
8295:
8291:
8287:
8277:
8274:
8270:
8266:
8262:
8258:
8254:
8247:
8243:
8239:
8232:
8228:
8221:
8217:
8209:
8205:
8201:
8193:
8189:
8185:
8181:
8173:
8169:
8165:
8161:
8155:
8151:
8147:
8143:
8136:
8132:
8128:
8124:
8117:
8113:
8109:
8105:
8101:
8097:
8090:
8086:
8079:
8072:
8068:
8064:
8056:
8052:
8023:
8015:
8011:
8002:
7998:
7989:
7985:
7976:
7972:
7966:
7962:
7956:
7952:
7948:
7940:
7936:
7927:
7923:
7914:
7910:
7901:
7897:
7891:
7887:
7881:
7877:
7873:
7867:
7864:
7861:
7855:
7852:
7845:
7844:
7843:
7838:
7834:
7829:
7825:
7820:
7816:
7811:
7807:
7800:
7796:
7791:
7787:
7782:
7778:
7774:
7769:
7765:
7761:
7731:
7727:
7718:
7714:
7705:
7701:
7694:
7686:
7682:
7676:
7672:
7666:
7662:
7658:
7653:
7649:
7644:
7638:
7634:
7627:
7619:
7615:
7609:
7605:
7600:
7594:
7590:
7584:
7580:
7576:
7570:
7561:
7558:
7555:
7545:
7544:
7543:
7531:
7527:
7523:
7518:
7514:
7506:
7499:
7495:
7491:
7486:
7482:
7474:
7467:
7463:
7456:
7452:
7445:
7441:
7434:
7430:
7415:
7411:
7404:
7400:
7396:
7389:
7385:
7376:
7366:
7362:
7358:
7354:
7348:
7344:
7340:
7333:
7325:
7321:
7317:
7313:
7307:
7304:
7300:
7294:
7290:
7286:
7282:
7276:
7268:
7261:
7257:
7253:
7240:
7238:
7236:
7231:
7229:
7228:superalgebras
7225:
7220:
7217:
7213:
7209:
7199:
7195:
7182:over a field
7181:
7178:is a general
7172:
7170:
7165:
7152:
7148:
7141:
7135:
7130:
7126:
7121:
7117:
7112:
7108:
7104:
7100:
7093:
7087:
7082:
7078:
7073:
7069:
7064:
7060:
7056:
7050:
7044:
7041:
7024:
7020:
7011:
7010:partial trace
7006:
6992:
6988:
6982:
6978:
6974:
6971:
6966:
6962:
6957:
6951:
6947:
6943:
6937:
6931:
6928:
6906:
6896:
6892:
6881:
6872:
6870:
6866:
6862:
6858:
6854:
6846:
6844:
6816:
6802:
6794:
6784:
6773:
6762:
6739:
6709:
6682:
6676:
6655:
6652:
6632:
6585:
6582:
6572:
6569:
6557:
6544:
6541:
6522:
6519:
6511:
6506:
6500:
6496:Two matrices
6494:
6490:
6457:
6444:
6425:
6411:
6408:
6405:
6391:
6372:
6369:
6360:
6347:
6328:
6325:
6311:
6294:
6292:
6266:
6253:
6242:
6231:
6212:
6209:
6186:
6183:
6169:
6166:
6160:
6157:
6154:
6143:
6132:
6123:
6117:
6112:
6111:bilinear form
6104:
6102:
6089:
6084:
6080:
6076:
6071:
6067:
6063:
6058:
6054:
6029:
6026:
6021:
6006:
6001:
5969:
5965:
5961:
5953:
5934:
5925:
5922:
5917:
5913:
5892:
5884:
5880:
5864:
5860:
5851:
5847:
5840:
5820:
5814:
5809:
5799:
5779:
5761:
5753:
5748:
5724:
5704:
5684:
5666:
5648:
5640:
5635:
5622:
5603:
5600:
5595:
5592:
5562:
5559:
5554:
5539:
5534:
5514:
5509:
5507:
5506:infinitesimal
5503:
5499:
5495:
5477:
5458:
5454:
5446:
5438:
5433:
5420:
5415:
5400:
5392:
5379:
5376:
5362:
5348:
5345:
5317:
5308:
5304:
5281:
5249:
5241:
5226:
5223:
5211:
5209:
5207:
5202:
5199:
5195:
5188:
5184:
5180:
5176:
5160:
5156:
5152:
5148:
5144:
5139:
5135:
5130:
5128:
5124:
5120:
5112:
5108:
5104:
5100:
5095:
5093:
5085:
5083:
5080:
5078:
5060:
5050:
5046:
5042:
5039:
5035:
5031:
5005:
5002:
4999:
4993:
4983:
4966:
4960:
4957:
4954:
4948:
4945:
4940:
4936:
4929:
4909:
4906:
4898:
4894:
4890:
4884:
4862:
4852:
4849:
4827:
4824:
4821:
4811:
4808:
4798:
4792:
4787:
4785:
4781:
4772:
4766:
4762:
4756:
4752:
4748:
4741:
4737:
4731:
4727:
4720:
4699:
4693:
4686:
4682:
4671:
4665:
4660:
4658:
4646:
4638:
4634:
4631:
4622:
4618:
4614:
4605:
4603:
4590:
4579:
4573:
4570:
4567:
4553:
4550:
4547:
4538:
4532:
4526:
4518:
4514:
4504:
4499:
4489:
4482:
4478:
4473:
4468:
4464:
4457:
4452:
4447:
4441:
4437:
4433:
4427:
4421:
4415:
4410:
4409:vector fields
4405:
4392:
4375:
4372:
4366:
4363:
4360:
4343:
4340:
4326:
4321:
4319:
4315:
4310:
4285:
4282:
4268:
4265:
4255:
4252:
4249:
4232:
4224:
4223:
4219:
4215:
4210:
4197:
4180:
4177:
4174:
4171:
4168:
4154:
4135:
4133:
4128:
4122:
4113:
4111:
4109:
4104:
4091:
4086:
4082:
4076:
4072:
4068:
4046:
4041:
4036:
4031:
4021:
4016:
4012:
4008:
4003:
3996:
3981:
3977:
3971:
3967:
3963:
3949:
3946:
3937:
3935:
3930:
3925:
3920:
3910:
3905:
3901:
3896:
3887:
3885:
3883:
3865:
3862:
3859:
3855:
3834:
3814:
3811:
3808:
3796:
3778:
3775:
3772:
3769:
3765:
3760:
3750:
3746:
3743:
3740:
3736:
3731:
3726:
3716:
3712:
3709:
3695:
3690:
3687:
3676:
3668:
3664:
3661:
3656:
3652:
3647:
3643:
3639:
3615:
3601:
3598:
3595:
3593:
3587:
3570:
3566:
3563:
3537:
3534:
3529:
3505:
3495:
3493:
3465:
3461:
3457:
3450:
3446:
3441:
3437:
3433:
3430:
3426:
3423:
3419:
3406:
3403:
3399:
3394:
3384:
3380:
3377:
3364:
3360:
3356:
3351:
3350:
3345:
3343:
3341:
3336:
3334:
3329:
3324:
3318:
3308:
3304:
3293:
3287:
3282:
3277:
3273:
3267:
3261:
3252:
3250:
3248:
3243:
3237:
3232:
3226:
3222:, ..., λ
3213:
3198:
3194:
3188:
3183:
3180:
3177:
3173:
3169:
3155:
3152:
3143:
3140:
3134:
3130:
3121:
3119:
3105:
3085:
3082:
3068:
3048:
3045:
3042:
3033:
3019:
2999:
2979:
2973:
2970:
2964:
2961:
2955:
2952:
2946:
2938:
2922:
2914:
2894:
2875:
2872:
2869:
2867:
2846:
2843:
2836:
2822:
2819:
2816:
2813:
2811:
2798:
2792:
2789:
2782:
2768:
2765:
2762:
2748:
2745:
2742:
2740:
2727:
2716:
2713:
2697:
2695:
2682:
2668:
2665:
2651:
2648:
2645:
2634:
2623:
2620:
2612:
2604:
2602:
2588:
2564:
2561:
2558:
2534:
2531:
2528:
2524:
2512:
2492:
2487:
2483:
2480:
2477:
2453:
2450:
2442:
2439:
2434:
2421:
2397:
2394:
2391:
2367:
2364:
2355:
2353:
2347:
2334:
2305:
2302:
2299:
2270:
2267:
2264:
2235:
2232:
2229:
2200:
2197:
2188:
2186:
2185:
2175:
2172:
2140:
2136:
2112:
2108:
2105:
2096:
2080:
2070:
2043:
2033:
2019:
2017:
2001:
1987:
1984:
1981:
1977:
1971:
1968:
1941:
1937:
1934:
1931:
1927:
1905:
1902:
1891:
1887:
1884:
1875:
1869:
1862:
1859:
1855:
1849:
1843:
1839:
1832:
1826:
1819:
1790:
1787:
1784:
1765:
1762:
1753:
1750:
1746:
1740:
1736:
1730:
1724:
1718:
1716:
1711:
1706:
1702:
1697:
1695:
1691:
1687:
1682:
1676:
1659:
1651:
1646:
1631:
1628:
1624:
1617:
1612:
1597:
1594:
1590:
1585:
1581:
1576:
1566:
1562:
1559:
1555:
1550:
1540:
1536:
1533:
1530:
1525:
1520:
1500:
1497:
1493:
1488:
1485:
1477:
1473:
1469:
1465:
1460:
1454:
1449:
1443:
1440:
1434:
1433:inner product
1428:
1425:
1421:
1418:
1411:
1404:
1401:
1395:
1390:
1384:
1379:
1378:vectorization
1370:
1366:
1360:
1347:
1341:
1338:
1334:
1328:
1325:
1321:
1315:
1310:
1307:
1304:
1300:
1294:
1289:
1286:
1283:
1279:
1275:
1271:
1247:
1243:
1240:
1237:
1233:
1209:
1205:
1202:
1199:
1195:
1171:
1167:
1164:
1161:
1157:
1133:
1129:
1126:
1117:
1113:
1107:
1101:
1096:
1088:
1086:
1083:
1070:
1066:
1049:
1045:
1042:
1039:
1025:
1022:
1014:
1009:
1004:
999:
993:
961:
958:
955:
952:
950:
937:
931:
928:
910:
907:
904:
890:
887:
884:
882:
869:
858:
855:
843:
835:
830:
828:
815:
812:
806:
803:
797:
794:
791:
788:
785:
780:
776:
772:
767:
763:
759:
754:
750:
746:
741:
738:
734:
728:
723:
720:
717:
713:
709:
695:
692:
683:
668:
662:
659:
654:
649:
642:
637:
632:
625:
620:
615:
609:
604:
599:
591:
587:
579:
575:
567:
563:
553:
549:
541:
537:
529:
525:
515:
511:
503:
499:
491:
487:
480:
475:
461:
452:
450:
445:
441:
437:
432:
426:
406:
384:
381:
377:
373:
370:
367:
362:
358:
354:
349:
345:
341:
336:
333:
329:
323:
318:
315:
312:
308:
304:
290:
287:
278:
274:
273:square matrix
270:
266:
261:
253:
251:
249:
245:
240:
238:
234:
230:
225:
219:
212:
208:
202:
197:
195:
191:
185:
178:
175:
171:
165:
160:
159:main diagonal
154:
147:
143:
142:square matrix
139:
135:
124:
121:
113:
110:November 2023
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
10108:Trace theory
10074:
10040:
10004:
9975:Hirsch, K.A.
9970:
9911:
9879:
9873:
9840:
9836:
9826:
9801:
9797:
9787:
9768:
9762:
9743:
9719:. Retrieved
9675:
9644:. Retrieved
9640:
9603:
9599:
9590:
9586:
9579:
9528:
8951:
8941:
8937:
8933:
8747:
8418:
8338:
8333:
8326:
8322:
8317:
8311:
8306:
8293:
8289:
8285:
8278:
8272:
8268:
8264:
8260:
8256:
8252:
8245:
8241:
8237:
8230:
8226:
8219:
8215:
8207:
8203:
8199:
8191:
8187:
8183:
8179:
8171:
8167:
8163:
8159:
8153:
8149:
8145:
8141:
8134:
8130:
8126:
8122:
8115:
8111:
8107:
8103:
8099:
8095:
8088:
8084:
8077:
8073:
8066:
8062:
8054:
8050:
8038:
7836:
7832:
7827:
7823:
7818:
7814:
7809:
7805:
7798:
7794:
7789:
7785:
7780:
7776:
7772:
7767:
7763:
7759:
7748:
7529:
7525:
7521:
7516:
7512:
7504:
7497:
7493:
7489:
7484:
7480:
7472:
7465:
7461:
7454:
7450:
7443:
7439:
7432:
7428:
7413:
7409:
7405:
7398:
7394:
7387:
7383:
7364:
7360:
7356:
7352:
7346:
7342:
7338:
7331:
7323:
7319:
7315:
7311:
7308:
7302:
7298:
7292:
7288:
7274:
7266:
7259:
7255:
7251:
7244:
7232:
7221:
7215:
7211:
7207:
7197:
7193:
7173:
7166:
7022:
7018:
7007:
6873:
6850:
6625:, such that
6558:
6509:
6504:
6498:
6495:
6488:
6458:
6361:
6295:
6291:Killing form
6121:
6115:
6108:
5750:In terms of
5749:
5636:
5510:
5505:
5448:
5440:
5434:
5306:
5302:
5215:
5203:
5197:
5193:
5186:
5182:
5178:
5174:
5158:
5154:
5150:
5146:
5142:
5131:
5122:
5118:
5110:
5096:
5089:
5086:Applications
5081:
4985:
4800:
4796:
4779:
4770:
4764:
4760:
4754:
4750:
4746:
4739:
4735:
4729:
4725:
4718:
4691:
4684:
4680:
4669:
4661:
4633:vector space
4628:is a finite-
4620:
4616:
4612:
4609:
4539:
4524:
4516:
4512:
4509:is given by
4497:
4487:
4480:
4476:
4469:
4462:
4455:
4445:
4439:
4435:
4431:
4425:
4419:
4413:
4406:
4322:
4314:differential
4311:
4225:
4211:
4136:
4130:denotes the
4126:
4120:
4117:
4105:
4044:
4039:
4026:
4019:
4001:
3998:
3939:
3928:
3915:
3908:
3894:
3891:
3800:
3693:
3685:
3674:
3670:
3654:
3650:
3634:
3455:
3453:is 1 if the
3448:
3444:
3440:fixed points
3370:
3358:
3354:
3337:
3327:
3322:
3317:Lie algebras
3306:
3302:
3291:
3285:
3275:
3271:
3265:
3259:
3256:
3241:
3235:
3224:
3215:
3145:
3138:
3132:
3128:
3125:
3034:
2701:
2608:
2437:
2435:
2356:
2351:
2349:
2190:
2181:
2179:
2098:
2020:
1873:
1867:
1860:
1857:
1853:
1841:
1837:
1830:
1824:
1821:
1755:
1748:
1744:
1738:
1734:
1728:
1722:
1719:
1709:
1698:
1680:
1674:
1464:vector space
1458:
1452:
1441:
1438:
1426:
1423:
1419:
1416:
1409:
1402:
1399:
1388:
1382:
1368:
1364:
1361:
1115:
1111:
1105:
1099:
1092:
1084:
1010:
997:
991:
839:
684:
459:
456:
436:real numbers
430:
424:
401:
276:
268:
264:
259:
257:
241:
237:vector space
223:
217:
210:
206:
198:
189:
183:
179:
173:
169:
163:
152:
145:
137:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
8955:Proof: Let
8389:Trace class
8369:Field trace
6853:trace class
6206:where
5494:Lie algebra
5212:Lie algebra
5163:of a group
5107:determinant
4630:dimensional
4218:determinant
4035:determinant
4013:, an upper
4011:Jordan form
3924:eigenvalues
3632:The matrix
3231:eigenvalues
3142:, there is
2187:, that is,
1394:dot product
844:. That is,
244:determinant
201:eigenvalues
10092:Categories
10037:Strang, G.
9997:Horn, R.A.
9954:0808.17003
9721:2020-09-09
9716:1079.00009
9646:2020-09-09
9615:References
8337:is called
7803:has trace
7470:such that
7224:supertrace
7192:tr :
5952:Lie groups
5754:, one has
5513:direct sum
5451:trace free
5134:characters
5123:loxodromic
4922:, we have
4698:dual space
4451:divergence
4214:derivative
4005:is always
3299:, because
3295:vanishes:
3281:commutator
3257:When both
3126:Given any
2018:as below.
1694:statistics
1001:, and all
831:Properties
422:column of
254:Definition
149:, denoted
80:newspapers
10081:EMS Press
10039:(2004) .
10003:(2013) .
9857:0004-5411
9818:0361-0918
9670:(2003) .
9446:
9411:
9390:
9361:⊕
9290:
9212:∑
9148:∑
9043:≠
8712:
8656:∑
8606:∑
8585:∑
8535:∑
8514:∑
8460:∑
8437:
7999:ψ
7973:φ
7963:∑
7953:∑
7924:φ
7898:ψ
7888:∑
7878:∑
7865:∘
7856:
7702:φ
7683:ψ
7673:∑
7663:∑
7616:ψ
7606:∑
7591:φ
7581:∑
7559:∘
7341:↦ φ(
7334:, φ)
7269:, φ)
7136:
7118:
7088:
7070:
7045:
6948:∑
6932:
6817:ρ
6803:ρ
6763:ϕ
6666:→
6653:ρ
6633:ρ
6570:ρ
6523:
6412:
6373:
6329:
6323:↦
6267:−
6213:
6187:
6170:
6161:
6085:∗
6077:×
6064:≠
6027:⊕
5929:∖
5918:∗
5890:→
5885:∗
5872:→
5857:→
5844:→
5818:→
5807:→
5785:→
5765:→
5710:→
5652:→
5604:
5588:↦
5560:⊕
5443:traceless
5401:∈
5349:
5247:→
5111:parabolic
5040:−
5032:±
4853:∈
4825:×
4812:∈
4574:
4554:
4376:
4367:
4344:
4283:⋅
4269:
4256:
4188:Δ
4181:
4169:≈
4159:Δ
4083:λ
4073:∏
3978:λ
3968:∑
3950:
3863:−
3812:×
3776:≡
3747:
3713:
3669:is zero.
3602:
3567:
3561:⟹
3535:−
3381:
3195:λ
3174:∑
3156:
3046:×
2876:
2847:
2823:
2793:
2769:
2749:
2717:
2669:
2652:
2635:⊗
2624:
2565:
2535:
2484:
2454:
2441:symmetric
2398:
2392:≠
2368:
2306:
2271:
2236:
2201:
2109:
2071:∈
2034:∈
1988:
1969:−
1938:
1903:−
1888:
1791:
1766:
1684:are real
1632:
1598:
1586:≤
1563:
1537:
1531:≤
1501:
1489:≤
1301:∑
1280:∑
1244:
1206:
1168:
1130:
1046:
1026:
1013:transpose
962:
932:
911:
891:
859:
804:−
714:∑
696:
660:−
420: th
413: th
371:⋯
309:∑
291:
190:traceless
9969:(1959).
8353:See also
8244:) → End(
8240:) ⊗ End(
8206:) → End(
8202:) × End(
7749:for any
7534:for any
7205:for all
7203:tr() = 0
6989:⟩
6958:⟨
5191:for all
5181:)) = tr(
5149: :
5119:elliptic
5117:, it is
4749: :
4712:and let
4689:, where
4615: :
4521:, where
4515:) · vol(
4503:net flow
4318:adjugate
3936:), then
3922:are the
3679:for all
3297:tr() = 0
3229:are the
1109:are two
415:row and
10083:, 2001
10027:2978290
9987:0107649
9946:1321145
9865:5827717
9708:1944431
9532:Proof:
8059:equals
7842:and so
7542:. Then
7412:) = tr(
7283:of the
7273:φ(
6113:(where
5905:(where
5683:scalars
5496:of the
4696:is the
4624:(where
4529:is the
4470:By the
4449:). Its
4216:of the
4045:product
4043:is the
4025:, ...,
4017:having
4009:to its
4007:similar
3914:, ...,
3904:complex
3880:in the
3827:matrix
3683:, then
3305:) = tr(
3136:matrix
1856:) = tr(
1713:by its
1703:on the
1422:) = tr(
1003:scalars
453:Example
434:can be
209:) = tr(
94:scholar
10051:
10025:
10015:
9985:
9952:
9944:
9934:
9886:
9863:
9855:
9816:
9775:
9750:
9714:
9706:
9696:
8192:φ
8184:ψ
8168:ψ
8160:φ
8154:ψ
8142:φ
8135:ψ
8127:φ
7824:φ
7806:ψ
7777:φ
7764:ψ
7513:ψ
7481:φ
7440:ψ
7429:φ
7359:→ Hom(
7318:→ Hom(
7279:. The
6871:norm.
5950:) for
5639:counit
5508:sets.
5125:. See
4716:be in
4708:be in
4704:. Let
4531:volume
4501:, the
3696:> 0
3218:λ
3216:where
3098:makes
1846:. The
1657:
399:where
262:of an
136:, the
96:
89:
82:
75:
67:
9861:S2CID
9593:) = 0
9566:is a
8410:Notes
8330:'
8297:'
8082:with
7511:) = Σ
7479:) = Σ
7369:. If
5171:) if
5115:[0,4)
4877:with
4645:basis
3677:) = 0
2992:then
2935:is a
2913:up to
2438:three
685:Then
444:field
260:trace
246:(see
140:of a
138:trace
101:JSTOR
87:books
10049:ISBN
10013:ISBN
9932:ISBN
9884:ISBN
9853:ISSN
9814:ISSN
9773:ISBN
9748:ISBN
9694:ISBN
9523:map.
9058:and
8940:)tr(
8236:End(
8198:End(
8139:to
8076:End(
8043:and
7502:and
7459:and
7437:and
7422:and
7393:Hom(
7030:and
7008:The
6502:and
6109:The
5437:zero
4678:and
4668:End(
4647:for
4523:vol(
4491:and
4454:div
4438:) =
3900:real
3660:rank
3599:rank
3311:and
3289:and
3269:are
3263:and
3035:For
3012:and
2058:and
1840:)tr(
1742:and
1732:are
1726:and
1692:and
1678:and
1468:norm
1456:and
1446:the
1386:and
1103:and
995:and
457:Let
258:The
221:and
177:).
73:news
9950:Zbl
9924:doi
9845:doi
9806:doi
9712:Zbl
9686:doi
9585:tr(
9379:as
8932:tr(
8309:⋅id
8061:tr(
8049:tr(
7753:in
7538:in
7408:tr(
7174:If
6874:If
6859:on
6855:of
6733:End
6724:on
6670:End
6512:if
5875:det
5447:or
5173:tr(
5136:of
5079:).
4700:of
4639:of
4533:of
4511:tr(
4461:tr(
4429:by
4423:on
4364:exp
4341:exp
4335:det
4266:adj
4236:det
4144:det
4118:If
4055:det
4037:of
3926:of
3902:or
3892:If
3673:tr(
3301:tr(
3283:of
3233:of
1852:tr(
1836:tr(
1672:if
1450:of
1437:tr(
1415:tr(
1398:tr(
250:).
205:tr(
182:tr(
151:tr(
132:In
56:by
10094::
10079:,
10073:,
10047:.
10023:MR
10021:.
10011:.
9999:;
9983:MR
9981:.
9948:.
9942:MR
9940:.
9930:.
9922:.
9914:.
9898:^
9859:.
9851:.
9841:58
9839:.
9835:.
9812:.
9802:18
9800:.
9796:.
9730:^
9710:.
9704:MR
9702:.
9692:.
9684:.
9655:^
9639:.
9622:^
9602:=
9443:tr
9408:tr
9387:tr
9287:tr
9278:11
9254:11
9105:11
8934:AB
8709:tr
8434:tr
8426::
8349:.
8332:→
8325:⊗
8292:⊗
8288:→
8271:⊗
8267:→
8263:⊗
8259:⊗
8255:⊗
8229:∘
8218:,
8190:,
8186:,
8182:,
8152:⊗
8133:,
8129:,
8125:,
8114:⊗
8110:→
8106:×
8102:×
8098:×
8087:⊗
8071:.
8065:∘
8053:∘
7853:tr
7762:↦
7414:BA
7410:AB
7397:,
7386:→
7363:,
7355:⊗
7322:,
7314:×
7306:.
7301:⊗
7291:⊗
7258:→
7254:×
7230:.
7222:A
7214:∈
7210:,
7196:↦
7171:.
7127:tr
7109:tr
7079:tr
7061:tr
7042:tr
7034::
7021:⊗
6929:tr
6790:tr
6705:tr
6545:0.
6520:tr
6409:tr
6370:tr
6326:tr
6210:ad
6184:ad
6167:ad
6158:tr
6119:,
6068:SL
6055:GL
5861:GL
5848:SL
5810:tr
5601:tr
5346:tr
5305:×
5224:tr
5208:.
5201:.
5196:∈
5189:))
5155:GL
5153:→
5145:,
5129:.
5094:.
4763:⊗
4753:→
4728:⊗
4683:⊗
4619:→
4571:tr
4551:tr
4537:.
4467:.
4440:Ax
4373:tr
4253:tr
4178:tr
4121:ΔA
4110:.
3947:tr
3744:tr
3710:tr
3653:=
3564:tr
3449:ii
3378:tr
3369:.
3357:×
3326:→
3320:gl
3313:tr
3307:BA
3303:AB
3274:×
3153:tr
3131:×
3020:tr
2873:tr
2844:tr
2820:tr
2790:tr
2766:tr
2746:tr
2714:tr
2666:tr
2649:tr
2621:tr
2562:tr
2532:tr
2481:tr
2451:tr
2395:tr
2365:tr
2354:.
2303:tr
2268:tr
2233:tr
2198:tr
2106:tr
1985:tr
1935:tr
1885:tr
1861:AP
1831:BA
1825:AB
1788:tr
1763:tr
1747:×
1737:×
1717:.
1696:.
1629:tr
1595:tr
1560:tr
1534:tr
1498:tr
1478::
1374:mn
1367:×
1241:tr
1203:tr
1165:tr
1127:tr
1114:×
1043:tr
1023:tr
1008:.
959:tr
929:tr
908:tr
888:tr
856:tr
781:33
768:22
755:11
693:tr
655:12
633:11
592:33
580:32
568:31
554:23
542:22
530:21
516:13
504:12
492:11
438:,
404:ii
363:22
350:11
288:tr
267:×
211:BA
207:AB
196:.
172:×
10057:.
10029:.
9989:.
9956:.
9926::
9892:.
9867:.
9847::
9820:.
9808::
9781:.
9756:.
9724:.
9688::
9649:.
9607:.
9604:0
9600:A
9591:A
9589:*
9587:A
9552:n
9546:l
9543:s
9510:,
9505:n
9499:l
9496:s
9473:0
9470:=
9467:)
9464:]
9461:B
9458:,
9455:A
9452:[
9449:(
9423:)
9420:A
9417:B
9414:(
9405:=
9402:)
9399:B
9396:A
9393:(
9367:,
9364:k
9356:n
9350:l
9347:s
9341:=
9336:n
9330:l
9327:g
9304:.
9301:)
9297:A
9293:(
9283:)
9274:e
9270:(
9266:f
9263:=
9259:)
9250:e
9246:(
9242:f
9237:i
9234:i
9230:]
9225:A
9221:[
9216:i
9208:=
9204:)
9199:j
9196:i
9192:e
9188:(
9184:f
9179:j
9176:i
9172:]
9167:A
9163:[
9158:j
9155:,
9152:i
9144:=
9141:)
9137:A
9133:(
9130:f
9110:)
9101:e
9097:(
9093:f
9090:=
9086:)
9081:j
9078:j
9074:e
9070:(
9066:f
9046:j
9040:i
9020:0
9017:=
9013:)
9008:j
9005:i
9001:e
8997:(
8993:f
8971:j
8968:i
8964:e
8946:.
8944:)
8942:B
8938:A
8918:,
8913:)
8907:0
8902:0
8895:0
8890:1
8884:(
8879:=
8875:B
8872:A
8851:,
8846:)
8840:0
8835:1
8828:0
8823:0
8817:(
8812:=
8808:B
8803:,
8798:)
8792:0
8787:0
8780:1
8775:0
8769:(
8764:=
8760:A
8731:.
8728:)
8724:A
8719:B
8715:(
8706:=
8701:j
8698:j
8693:)
8688:A
8683:B
8678:(
8671:n
8666:1
8663:=
8660:j
8652:=
8647:j
8644:i
8640:a
8634:i
8631:j
8627:b
8621:m
8616:1
8613:=
8610:i
8600:n
8595:1
8592:=
8589:j
8581:=
8576:i
8573:j
8569:b
8563:j
8560:i
8556:a
8550:n
8545:1
8542:=
8539:j
8529:m
8524:1
8521:=
8518:i
8510:=
8505:i
8502:i
8497:)
8492:B
8487:A
8482:(
8475:m
8470:1
8467:=
8464:i
8456:=
8453:)
8449:B
8444:A
8440:(
8334:F
8327:V
8323:V
8312:V
8307:c
8302:c
8294:V
8290:V
8286:F
8281:V
8273:V
8269:V
8265:V
8261:V
8257:V
8253:V
8248:)
8246:V
8242:V
8238:V
8231:g
8227:f
8222:)
8220:g
8216:f
8214:(
8210:)
8208:V
8204:V
8200:V
8194:)
8188:v
8180:w
8178:(
8174:)
8172:v
8170:(
8166:)
8164:w
8162:(
8150:v
8148:)
8146:w
8144:(
8137:)
8131:w
8123:v
8121:(
8116:V
8112:V
8108:V
8104:V
8100:V
8096:V
8089:V
8085:V
8080:)
8078:V
8069:)
8067:S
8063:T
8057:)
8055:T
8051:S
8045:T
8041:S
8024:.
8021:)
8016:i
8012:v
8008:(
8003:j
7995:)
7990:j
7986:w
7982:(
7977:i
7967:i
7957:j
7949:=
7946:)
7941:j
7937:w
7933:(
7928:i
7920:)
7915:i
7911:v
7907:(
7902:j
7892:j
7882:i
7874:=
7871:)
7868:T
7862:S
7859:(
7840:)
7837:j
7833:w
7831:(
7828:i
7822:)
7819:i
7815:v
7813:(
7810:j
7799:i
7795:v
7793:)
7790:j
7786:w
7784:(
7781:i
7775:)
7773:u
7771:(
7768:j
7760:u
7755:V
7751:u
7732:i
7728:v
7724:)
7719:j
7715:w
7711:(
7706:i
7698:)
7695:u
7692:(
7687:j
7677:j
7667:i
7659:=
7654:i
7650:v
7645:)
7639:j
7635:w
7631:)
7628:u
7625:(
7620:j
7610:j
7601:(
7595:i
7585:i
7577:=
7574:)
7571:u
7568:(
7565:)
7562:T
7556:S
7553:(
7540:V
7536:u
7530:j
7526:w
7524:)
7522:u
7520:(
7517:j
7509:u
7507:(
7505:T
7498:i
7494:v
7492:)
7490:u
7488:(
7485:i
7477:u
7475:(
7473:S
7466:j
7462:w
7455:i
7451:v
7444:j
7433:i
7424:T
7420:S
7416:)
7401:)
7399:V
7395:V
7388:V
7384:V
7379:V
7371:V
7367:)
7365:V
7361:V
7357:V
7353:V
7347:v
7345:)
7343:w
7339:w
7332:v
7330:(
7326:)
7324:V
7320:V
7316:V
7312:V
7303:V
7299:V
7293:V
7289:V
7277:)
7275:v
7267:v
7265:(
7260:F
7256:V
7252:V
7247:V
7216:A
7212:b
7208:a
7198:k
7194:A
7188:A
7184:k
7176:A
7153:.
7149:)
7145:)
7142:Z
7139:(
7131:A
7122:(
7113:B
7105:=
7101:)
7097:)
7094:Z
7091:(
7083:B
7074:(
7065:A
7057:=
7054:)
7051:Z
7048:(
7032:B
7028:A
7023:B
7019:A
7014:Z
6993:,
6983:n
6979:e
6975:K
6972:,
6967:n
6963:e
6952:n
6944:=
6941:)
6938:K
6935:(
6907:n
6903:)
6897:n
6893:e
6889:(
6876:K
6831:)
6828:)
6824:Y
6820:(
6814:)
6810:X
6806:(
6800:(
6795:V
6785:=
6782:)
6778:Y
6774:,
6770:X
6766:(
6743:)
6740:V
6737:(
6710:V
6683:.
6680:)
6677:V
6674:(
6661:g
6656::
6611:g
6589:)
6586:V
6583:,
6578:g
6573:,
6567:(
6542:=
6539:)
6535:Y
6530:X
6526:(
6505:Y
6499:X
6489:n
6473:l
6470:s
6445:.
6442:)
6438:Z
6434:]
6430:Y
6426:,
6422:X
6418:[
6415:(
6406:=
6403:)
6400:]
6396:Z
6392:,
6388:Y
6384:[
6380:X
6376:(
6348:.
6345:)
6341:Y
6336:X
6332:(
6320:)
6316:Y
6312:,
6308:X
6304:(
6276:X
6271:Y
6263:Y
6258:X
6254:=
6251:]
6247:Y
6243:,
6239:X
6235:[
6232:=
6228:Y
6224:)
6220:X
6216:(
6201:)
6198:)
6194:Y
6190:(
6181:)
6177:X
6173:(
6164:(
6155:=
6152:)
6148:Y
6144:,
6140:X
6136:(
6133:B
6122:Y
6116:X
6090:.
6081:K
6072:n
6059:n
6044:n
6030:K
6022:n
6016:l
6013:s
6007:=
6002:n
5996:l
5993:g
5970:n
5966:/
5962:1
5938:}
5935:0
5932:{
5926:K
5923:=
5914:K
5893:1
5881:K
5865:n
5852:n
5841:1
5821:0
5815:K
5800:n
5794:l
5791:g
5780:n
5774:l
5771:s
5762:0
5745:n
5741:n
5725:n
5719:l
5716:g
5705:n
5699:l
5696:g
5667:n
5661:l
5658:g
5649:K
5623:.
5619:I
5615:)
5611:A
5607:(
5596:n
5593:1
5584:A
5563:K
5555:n
5549:l
5546:s
5540:=
5535:n
5529:l
5526:g
5478:n
5472:l
5469:s
5421:.
5416:n
5410:l
5407:g
5397:B
5393:,
5389:A
5380:0
5377:=
5374:)
5371:]
5367:B
5363:,
5359:A
5355:[
5352:(
5336:K
5332:K
5318:K
5307:n
5303:n
5298:n
5282:n
5276:l
5273:g
5250:K
5242:n
5236:l
5233:g
5227::
5198:G
5194:g
5187:g
5185:(
5183:B
5179:g
5177:(
5175:A
5169:V
5165:G
5161:)
5159:V
5157:(
5151:G
5147:B
5143:A
5075:(
5061:n
5057:}
5051:2
5047:/
5043:1
5036:n
5029:{
5009:)
5006:I
5003:,
5000:0
4997:(
4994:N
4970:)
4967:W
4964:(
4961:r
4958:t
4955:=
4952:]
4949:u
4946:W
4941:T
4937:u
4933:[
4930:E
4910:I
4907:=
4904:]
4899:T
4895:u
4891:u
4888:[
4885:E
4863:n
4858:R
4850:u
4828:n
4822:n
4817:R
4809:W
4782:*
4780:V
4775:V
4771:f
4767:*
4765:V
4761:V
4755:V
4751:V
4747:f
4742:)
4740:v
4738:(
4736:g
4730:g
4726:v
4721:*
4719:V
4714:g
4710:V
4706:v
4702:V
4694:*
4692:V
4687:*
4685:V
4681:V
4676:V
4672:)
4670:V
4653:f
4649:V
4641:f
4626:V
4621:V
4617:V
4613:f
4591:.
4588:)
4584:X
4580:d
4577:(
4568:=
4565:)
4561:X
4557:(
4548:d
4535:U
4527:)
4525:U
4519:)
4517:U
4513:A
4507:U
4498:R
4493:U
4488:x
4483:)
4481:x
4479:(
4477:F
4465:)
4463:A
4456:F
4446:A
4436:x
4434:(
4432:F
4426:R
4420:F
4414:A
4393:.
4390:)
4387:)
4383:A
4379:(
4370:(
4361:=
4358:)
4355:)
4351:A
4347:(
4338:(
4296:)
4290:A
4286:d
4280:)
4276:A
4272:(
4261:(
4250:=
4247:)
4243:A
4239:(
4233:d
4198:.
4195:)
4191:A
4184:(
4175:+
4172:1
4166:)
4162:A
4155:+
4151:I
4147:(
4127:I
4092:.
4087:i
4077:i
4069:=
4066:)
4062:A
4058:(
4040:A
4029:n
4027:λ
4023:1
4020:λ
4002:A
3982:i
3972:i
3964:=
3961:)
3957:A
3953:(
3929:A
3918:n
3916:λ
3912:1
3909:λ
3895:A
3866:1
3860:n
3856:t
3835:A
3815:n
3809:n
3779:0
3773:n
3770:=
3766:)
3761:n
3756:I
3751:(
3741:=
3737:)
3732:k
3727:n
3722:I
3717:(
3700:n
3694:n
3686:A
3681:k
3675:A
3662:.
3655:A
3651:A
3637:X
3635:P
3616:.
3613:)
3609:X
3605:(
3596:=
3588:)
3582:X
3576:P
3571:(
3551:T
3545:X
3538:1
3530:)
3525:X
3518:T
3512:X
3506:(
3500:X
3496:=
3486:X
3480:P
3456:i
3445:a
3424:.
3407:n
3404:=
3400:)
3395:n
3390:I
3385:(
3367:n
3359:n
3355:n
3328:k
3323:n
3309:)
3292:B
3286:A
3276:n
3272:n
3266:B
3260:A
3242:A
3236:A
3225:n
3220:1
3199:i
3189:n
3184:1
3181:=
3178:i
3170:=
3167:)
3163:A
3159:(
3139:A
3133:n
3129:n
3106:f
3086:n
3083:=
3080:)
3076:I
3072:(
3069:f
3049:n
3043:n
3000:f
2980:,
2977:)
2974:x
2971:y
2968:(
2965:f
2962:=
2959:)
2956:y
2953:x
2950:(
2947:f
2923:f
2895:,
2892:)
2888:A
2883:B
2879:(
2870:=
2863:)
2859:B
2854:A
2850:(
2837:,
2834:)
2830:A
2826:(
2817:c
2814:=
2807:)
2803:A
2799:c
2796:(
2783:,
2780:)
2776:B
2772:(
2763:+
2760:)
2756:A
2752:(
2743:=
2736:)
2732:B
2728:+
2724:A
2720:(
2683:.
2680:)
2676:B
2672:(
2663:)
2659:A
2655:(
2646:=
2643:)
2639:B
2631:A
2627:(
2589:,
2586:)
2582:B
2577:C
2572:A
2568:(
2559:=
2556:)
2552:A
2547:B
2542:C
2538:(
2529:=
2525:)
2519:T
2513:)
2508:C
2503:B
2498:A
2493:(
2488:(
2478:=
2475:)
2471:C
2466:B
2461:A
2457:(
2422:.
2419:)
2415:B
2410:C
2405:A
2401:(
2389:)
2385:C
2380:B
2375:A
2371:(
2335:.
2332:)
2328:C
2323:B
2318:A
2313:D
2309:(
2300:=
2297:)
2293:B
2288:A
2283:D
2278:C
2274:(
2265:=
2262:)
2258:A
2253:D
2248:C
2243:B
2239:(
2230:=
2227:)
2223:D
2218:C
2213:B
2208:A
2204:(
2159:b
2152:T
2146:a
2141:=
2137:)
2130:T
2124:a
2118:b
2113:(
2081:n
2076:R
2067:b
2044:n
2039:R
2030:a
2002:.
1999:)
1995:A
1991:(
1982:=
1978:)
1972:1
1964:P
1959:)
1955:P
1950:A
1946:(
1942:(
1932:=
1928:)
1924:)
1920:P
1915:A
1911:(
1906:1
1898:P
1892:(
1874:P
1868:A
1863:)
1858:P
1854:A
1844:)
1842:B
1838:A
1807:)
1803:A
1798:B
1794:(
1785:=
1782:)
1778:B
1773:A
1769:(
1749:m
1745:n
1739:n
1735:m
1729:B
1723:A
1710:B
1681:B
1675:A
1660:,
1652:2
1647:]
1643:)
1639:B
1635:(
1625:[
1618:2
1613:]
1609:)
1605:A
1601:(
1591:[
1582:)
1577:2
1572:B
1567:(
1556:)
1551:2
1546:A
1541:(
1526:2
1521:]
1517:)
1513:B
1508:A
1504:(
1494:[
1486:0
1459:B
1453:A
1444:)
1442:B
1439:A
1429:)
1427:A
1424:B
1420:B
1417:A
1410:A
1405:)
1403:A
1400:A
1389:B
1383:A
1369:n
1365:m
1348:.
1342:j
1339:i
1335:b
1329:j
1326:i
1322:a
1316:n
1311:1
1308:=
1305:j
1295:m
1290:1
1287:=
1284:i
1276:=
1272:)
1265:T
1259:A
1253:B
1248:(
1238:=
1234:)
1229:A
1222:T
1216:B
1210:(
1200:=
1196:)
1189:T
1183:B
1177:A
1172:(
1162:=
1158:)
1153:B
1146:T
1140:A
1134:(
1116:n
1112:m
1106:B
1100:A
1071:.
1067:)
1061:T
1055:A
1050:(
1040:=
1037:)
1033:A
1029:(
1006:c
998:B
992:A
973:)
969:A
965:(
956:c
953:=
946:)
942:A
938:c
935:(
922:)
918:B
914:(
905:+
902:)
898:A
894:(
885:=
878:)
874:B
870:+
866:A
862:(
816:1
813:=
810:)
807:5
801:(
798:+
795:5
792:+
789:1
786:=
777:a
773:+
764:a
760:+
751:a
747:=
742:i
739:i
735:a
729:3
724:1
721:=
718:i
710:=
707:)
703:A
699:(
669:)
663:5
650:6
643:2
638:5
626:3
621:0
616:1
610:(
605:=
600:)
588:a
576:a
564:a
550:a
538:a
526:a
512:a
500:a
488:a
481:(
476:=
472:A
460:A
447:F
431:A
425:A
418:i
411:i
402:a
385:n
382:n
378:a
374:+
368:+
359:a
355:+
346:a
342:=
337:i
334:i
330:a
324:n
319:1
316:=
313:i
305:=
302:)
298:A
294:(
277:A
269:n
265:n
224:B
218:A
213:)
186:)
184:A
174:n
170:n
164:A
155:)
153:A
146:A
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
