4356:
5975:
1479:
380:
1176:
651:
2107:
128:
1474:{\displaystyle {\begin{matrix}\ell _{1,1}x_{1}&&&&&&&=&b_{1}\\\ell _{2,1}x_{1}&+&\ell _{2,2}x_{2}&&&&&=&b_{2}\\\vdots &&\vdots &&\ddots &&&&\vdots \\\ell _{m,1}x_{1}&+&\ell _{m,2}x_{2}&+&\dotsb &+&\ell _{m,m}x_{m}&=&b_{m}\\\end{matrix}}}
3260:
2938:
399:
1826:
3590:
375:{\displaystyle L={\begin{bmatrix}\ell _{1,1}&&&&0\\\ell _{2,1}&\ell _{2,2}&&&\\\ell _{3,1}&\ell _{3,2}&\ddots &&\\\vdots &\vdots &\ddots &\ddots &\\\ell _{n,1}&\ell _{n,2}&\ldots &\ell _{n,n-1}&\ell _{n,n}\end{bmatrix}}}
4459:
throughout; in particular the lower triangular matrices also form a Lie algebra. However, operations mixing upper and lower triangular matrices do not in general produce triangular matrices. For instance, the sum of an upper and a lower triangular matrix can be any matrix; the product of a lower
3091:
2769:
4021:-dimensional affine space, and the existence of a (common) eigenvalue (and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz. In algebraic terms, these operators correspond to an
646:{\displaystyle U={\begin{bmatrix}u_{1,1}&u_{1,2}&u_{1,3}&\ldots &u_{1,n}\\&u_{2,2}&u_{2,3}&\ldots &u_{2,n}\\&&\ddots &\ddots &\vdots \\&&&\ddots &u_{n-1,n}\\0&&&&u_{n,n}\end{bmatrix}}}
3895:
are simultaneously triangularizable. This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before. This was proven by
Frobenius, starting in 1878 for a commuting pair, as discussed at
2102:{\displaystyle {\begin{aligned}x_{1}&={\frac {b_{1}}{\ell _{1,1}}},\\x_{2}&={\frac {b_{2}-\ell _{2,1}x_{1}}{\ell _{2,2}}},\\&\ \ \vdots \\x_{m}&={\frac {b_{m}-\sum _{i=1}^{m-1}\ell _{m,i}x_{i}}{\ell _{m,m}}}.\end{aligned}}}
787:
873:
3454:
4613:, any finite-dimensional nilpotent Lie algebra is conjugate to a subalgebra of the strictly upper triangular matrices, that is to say, a finite-dimensional nilpotent Lie algebra is simultaneously strictly upper triangularizable.
4694:
components accordingly as each diagonal entry is positive or negative. The identity component is invertible triangular matrices with positive entries on the diagonal, and the group of all invertible triangular matrices is a
4269:
the commutator vanishes so this holds. This was proven by Drazin, Dungey, and
Gruenberg in 1951; a brief proof is given by Prasolov in 1994. One direction is clear: if the matrices are simultaneously triangularisable, then
2406:
3255:{\displaystyle A={\begin{bmatrix}A_{11}&0&\cdots &0\\A_{21}&A_{22}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\A_{k1}&A_{k2}&\cdots &A_{kk}\end{bmatrix}}}
2933:{\displaystyle A={\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1k}\\0&A_{22}&\cdots &A_{2k}\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &A_{kk}\end{bmatrix}}}
3329:
3007:
3634:
is similar to an upper triangular matrix of a very particular form. The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.
2574:
1831:
4790:. The invertible ones among them form a subgroup of the general linear group, whose conjugate subgroups are those defined as the stabilizer of some (other) complete flag. These subgroups are
3813:
Simultaneous triangularizability means that this algebra is conjugate into the Lie subalgebra of upper triangular matrices, and is equivalent to this algebra being a Lie subalgebra of a
950:
917:
4175:
1538:
3449:
1818:
4091:
3893:
3711:
2276:
1739:
4580:
4499:
4775:
4604:
4527:
3373:
3051:
4801:
all triangular matrices). The conjugates of such a group are the subgroups defined as the stabilizer of some partial flag. These subgroups are called parabolic subgroups.
4797:
The stabilizer of a partial flag obtained by forgetting some parts of the standard flag can be described as a set of block upper triangular matrices (but its elements are
4794:. The group of invertible lower triangular matrices is such a subgroup, since it is the stabilizer of the standard flag associated to the standard basis in reverse order.
5633:
2485:
1119:
3753:
4724:
4692:
4345:
4267:
1766:
1673:
1646:
1619:
1592:
1565:
1146:
1078:
1047:
1020:
985:
694:
A non-square (or sometimes any) matrix with zeros above (below) the diagonal is called a lower (upper) trapezoidal matrix. The non-zero entries form the shape of a
3585:{\displaystyle 0<\left\langle e_{1}\right\rangle <\left\langle e_{1},e_{2}\right\rangle <\cdots <\left\langle e_{1},\ldots ,e_{n}\right\rangle =K^{n}.}
2447:
712:
3847:
3811:
798:
4015:
3960:
3079:
2757:
1693:
4314:
4236:
5847:
4355:
5066:
5938:
4423:
of square matrices for a given size. Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the
2212:
3721:
if there is a basis under which they are all upper triangular; equivalently, if they are upper triangularizable by a single similarity matrix
4931:
5857:
5623:
4733:
of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a
6011:
5024:
4958:
4895:
3725:
Such a set of matrices is more easily understood by considering the algebra of matrices it generates, namely all polynomials in the
2300:
3269:
2947:
5658:
5205:
4664:
of all invertible matrices. A triangular matrix is invertible precisely when its diagonal entries are invertible (non-zero).
2490:
3662:
5422:
5059:
4387:
3904:
3596:
acts transitively on bases), so any matrix that stabilises a flag is similar to one that stabilizes the standard flag.
2672:
5497:
3637:
In the case of complex matrices, it is possible to say more about triangularization, namely, that any square matrix
2652:
If all of the entries on the main diagonal of a (upper or lower) triangular matrix are also 0, the matrix is called
5653:
5175:
3612:
960:
for upper triangular matrices. The process is so called because for lower triangular matrices, one first computes
5757:
5628:
5542:
2216:
922:
889:
5862:
5752:
5460:
5140:
684:
4818:
5897:
5826:
5708:
5568:
5165:
5052:
4096:
2189:
2125:
1487:
3653:
as change of basis) to an upper triangular matrix; this follows by taking an
Hermitian basis for the flag.
3402:
2192:
of a triangular matrix equal the product of the diagonal entries, as can be checked by direct computation.
5767:
5350:
5155:
4844:
4022:
2208:
1771:
111:
38:
4050:
3852:
3670:
2221:
5713:
5450:
5300:
5295:
5130:
5105:
5100:
4787:
4471:
3392:
1698:
96:
5974:
4532:
2147:
A matrix which is both symmetric and triangular is diagonal. In a similar vein, a matrix which is both
4817:
of the field of scalars; in the case of complex numbers it corresponds to a group formed of parabolic
4477:
5907:
5265:
5095:
5075:
4834:
4756:
4734:
4661:
4585:
4508:
4441:
4037:
3593:
3334:
3012:
2698:
3900:. As for a single matrix, over the complex numbers these can be triangularized by unitary matrices.
5928:
5902:
5480:
5285:
5275:
4653:
4617:
4502:
4420:
4041:
3642:
3604:
2167:
2199:
of a triangular matrix are exactly its diagonal entries. Moreover, each eigenvalue occurs exactly
83:
Because matrix equations with triangular matrices are easier to solve, they are very important in
5979:
5933:
5923:
5877:
5872:
5801:
5737:
5603:
5340:
5335:
5270:
5260:
5125:
4859:
4854:
4696:
3897:
3821:
3627:
84:
4610:
2452:
3619:
has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that
6016:
5990:
5777:
5772:
5762:
5742:
5703:
5698:
5527:
5522:
5507:
5502:
5493:
5488:
5435:
5280:
5225:
5195:
5190:
5170:
5160:
5120:
5030:
5020:
4964:
4954:
4927:
4901:
4891:
4849:
92:
4983:
4347:
or combination thereof – it will still have 0s on the diagonal in the triangularizing basis.
3903:
The fact that commuting matrices have a common eigenvector can be interpreted as a result of
1091:
782:{\displaystyle {\begin{bmatrix}1&0&0\\2&96&0\\4&9&69\\\end{bmatrix}}}
5985:
5953:
5882:
5821:
5816:
5796:
5732:
5638:
5608:
5593:
5578:
5573:
5512:
5465:
5440:
5430:
5401:
5320:
5315:
5290:
5220:
5200:
5110:
5090:
4995:
4839:
4822:
4751:
4647:
4445:
4033:
3814:
3728:
3615:) is similar to a triangular matrix. This can be proven by using induction on the fact that
2684:
2664:
88:
31:
4706:
4670:
4323:
4245:
1744:
1651:
1624:
1597:
1570:
1543:
1124:
1056:
1025:
998:
963:
868:{\displaystyle {\begin{bmatrix}1&4&1\\0&6&9\\0&0&1\\\end{bmatrix}}}
5683:
5618:
5598:
5583:
5563:
5547:
5445:
5376:
5366:
5325:
5210:
5180:
4700:
4363:
2625:
2423:
2170:) and triangular is also diagonal. This can be seen by looking at the diagonal entries of
680:
3826:
2593:
of a (upper or lower) triangular matrix are all 1, the matrix is called (upper or lower)
3758:
5943:
5887:
5867:
5852:
5811:
5688:
5648:
5613:
5537:
5476:
5455:
5396:
5386:
5371:
5305:
5250:
5240:
5235:
5145:
4791:
4738:
4643:
4405:
4359:
3965:
3910:
3650:
3623:
stabilizes a flag, and is thus triangularizable with respect to a basis for that flag.
3384:
3064:
2742:
1678:
107:
17:
4320:
upper triangularizable (hence nilpotent), which is preserved by multiplication by any
4273:
4195:
6005:
5948:
5806:
5747:
5678:
5668:
5663:
5588:
5517:
5391:
5381:
5310:
5230:
5215:
5150:
4814:
4625:
3396:
2590:
2148:
65:
49:
2701:
are zero, except for the entries in a single column. Such a matrix is also called a
5831:
5788:
5693:
5406:
5345:
5255:
5135:
5019:. Simeon Ivanov. Providence, R.I.: American Mathematical Society. p. 178–179.
4652:
The set of invertible triangular matrices of a given kind (upper or lower) forms a
4632:
4621:
4380:
4376:
4367:
4040:
is simultaneously upper triangularizable, the case of commuting matrices being the
2728:
2722:
4460:
triangular with an upper triangular matrix is not necessarily triangular either.
5673:
5643:
5411:
5245:
5115:
4730:
4428:
4424:
2634:
2185:
2129:
668:, and an upper or right triangular matrix is commonly denoted with the variable
4999:
5724:
5185:
4810:
4432:
4416:
4239:
2196:
1648:, and thus can be solved once one substitutes in the already solved value for
5958:
5532:
5034:
4905:
4657:
4464:
4178:
2641:
2141:
695:
4968:
4411:
The product of an upper triangular matrix and a scalar is upper triangular.
2144:
of an upper triangular matrix is a lower triangular matrix and vice versa.
664:. A lower or left triangular matrix is commonly denoted with the variable
5892:
5014:
4885:
4948:
4616:
Algebras of upper triangular matrices have a natural generalization in
4926:(2 ed.). Princeton, NJ: Princeton University Press. p. 168.
4786:
The upper triangular matrices are precisely those that stabilize the
4415:
Together these facts mean that the upper triangular matrices form a
3820:
The basic result is that (over an algebraically closed field), the
4354:
3395:: upper triangular matrices are precisely those that preserve the
5044:
4470:
The set of strictly upper (or lower) triangular matrices forms a
4408:
of an upper triangular matrix, if it exists, is upper triangular.
4401:
The product of two upper triangular matrices is upper triangular.
3599:
Any complex square matrix is triangularizable. In fact, a matrix
4529:, the Lie algebra of all upper triangular matrices; in symbols,
2282:. In other words, the characteristic polynomial of a triangular
5048:
4606:
is the Lie algebra of the Lie group of unitriangular matrices.
2401:{\displaystyle p_{A}(x)=(x-a_{11})(x-a_{22})\cdots (x-a_{nn})}
4398:
The sum of two upper triangular matrices is upper triangular.
4093:
is simultaneously triangularisable if and only if the matrix
2697:
is a special form of unitriangular matrix, where all of the
2116:
can be solved in an analogous way, only working backwards.
4667:
Over the real numbers, this group is disconnected, having
3324:{\displaystyle A_{ij}\in \mathbb {F} ^{n_{i}\times n_{j}}}
3002:{\displaystyle A_{ij}\in \mathbb {F} ^{n_{i}\times n_{j}}}
4890:(2nd ed.). New York: Springer. pp. 86–87, 169.
4440:. The Lie algebra of all upper triangular matrices is a
1151:
Notice that this does not require inverting the matrix.
4982:
Drazin, M. P.; Dungey, J. W.; Gruenberg, K. W. (1951).
2633:
triangular matrix has nothing to do with the notion of
4394:
Upper triangularity is preserved by many operations:
3106:
2784:
1181:
807:
721:
414:
143:
4759:
4709:
4673:
4588:
4535:
4511:
4480:
4326:
4276:
4248:
4198:
4099:
4053:
3968:
3913:
3855:
3829:
3761:
3731:
3673:
3457:
3405:
3337:
3272:
3094:
3067:
3015:
2950:
2772:
2745:
2569:{\displaystyle (x-a_{11})(x-a_{22})\cdots (x-a_{nn})}
2493:
2455:
2426:
2303:
2224:
1829:
1774:
1747:
1701:
1681:
1654:
1627:
1600:
1573:
1546:
1490:
1179:
1127:
1094:
1059:
1028:
1001:
966:
952:
is very easy to solve by an iterative process called
925:
892:
801:
715:
402:
131:
679:
A matrix that is both upper and lower triangular is
5916:
5840:
5786:
5722:
5556:
5474:
5420:
5359:
5083:
4953:(2nd ed.). New York: Wiley. pp. 285–290.
2416:polynomial whose roots are the diagonal entries of
4769:
4726:on the diagonal, corresponding to the components.
4718:
4686:
4598:
4574:
4521:
4493:
4339:
4308:
4261:
4230:
4169:
4085:
4009:
3954:
3887:
3841:
3805:
3747:
3705:
3584:
3443:
3391:. Abstractly, this is equivalent to stabilizing a
3367:
3323:
3254:
3073:
3045:
3001:
2932:
2751:
2731:(partitioned matrix) that is a triangular matrix.
2568:
2479:
2441:
2400:
2270:
2112:A matrix equation with an upper triangular matrix
2101:
1812:
1760:
1733:
1687:
1667:
1640:
1613:
1586:
1559:
1532:
1473:
1140:
1113:
1072:
1041:
1014:
979:
944:
911:
867:
781:
645:
374:
4809:The group of 2×2 upper unitriangular matrices is
2420:(with multiplicities). To see this, observe that
4821:; the 3×3 upper unitriangular matrices form the
4047:More generally and precisely, a set of matrices
3907:: commuting matrices form a commutative algebra
2456:
2247:
4924:Matrix mathematics: theory, facts, and formulas
4917:
4915:
3649:is unitarily equivalent (i.e. similar, using a
3399:, which is given by the standard ordered basis
1170:can be written as a system of linear equations
68:are zero. Similarly, a square matrix is called
4427:of square matrices of a fixed size, where the
3630:theorem, which states that in this situation,
956:for lower triangular matrices and analogously
5060:
2449:is also triangular and hence its determinant
8:
2663:All finite strictly triangular matrices are
1594:directly. The second equation only involves
4448:of the Lie algebra of all square matrices.
4036:, which shows that any representation of a
1049:. In an upper triangular matrix, one works
5634:Fundamental (linear differential equation)
5067:
5053:
5045:
4988:Journal of the London Mathematical Society
4463:The set of unitriangular matrices forms a
2124:Forward substitution is used in financial
945:{\displaystyle U\mathbf {x} =\mathbf {b} }
912:{\displaystyle L\mathbf {x} =\mathbf {b} }
4761:
4760:
4758:
4708:
4678:
4672:
4590:
4589:
4587:
4560:
4559:
4550:
4549:
4537:
4536:
4534:
4513:
4512:
4510:
4482:
4481:
4479:
4388:powers of the 4-bit Gray code permutation
4331:
4325:
4297:
4284:
4275:
4253:
4247:
4219:
4206:
4197:
4158:
4145:
4129:
4110:
4098:
4077:
4058:
4052:
4044:case, abelian being a fortiori solvable.
4017:which can be interpreted as a variety in
3998:
3979:
3967:
3943:
3924:
3912:
3879:
3860:
3854:
3828:
3791:
3772:
3760:
3736:
3730:
3697:
3678:
3672:
3626:A more precise statement is given by the
3573:
3555:
3536:
3507:
3494:
3472:
3456:
3432:
3413:
3404:
3387:to a triangular matrix is referred to as
3336:
3313:
3300:
3295:
3291:
3290:
3277:
3271:
3235:
3215:
3200:
3154:
3142:
3113:
3101:
3093:
3066:
3014:
2991:
2978:
2973:
2969:
2968:
2955:
2949:
2913:
2859:
2842:
2820:
2803:
2791:
2779:
2771:
2744:
2554:
2529:
2507:
2492:
2454:
2425:
2386:
2361:
2339:
2308:
2302:
2229:
2223:
2078:
2067:
2051:
2035:
2024:
2011:
2004:
1991:
1952:
1941:
1925:
1912:
1905:
1892:
1867:
1857:
1851:
1838:
1830:
1828:
1798:
1779:
1773:
1752:
1746:
1725:
1706:
1700:
1680:
1659:
1653:
1632:
1626:
1605:
1599:
1578:
1572:
1551:
1545:
1524:
1511:
1495:
1489:
1461:
1444:
1428:
1401:
1385:
1368:
1352:
1311:
1290:
1274:
1257:
1241:
1227:
1204:
1188:
1180:
1178:
1132:
1126:
1099:
1093:
1064:
1058:
1033:
1027:
1006:
1000:
971:
965:
937:
929:
924:
904:
896:
891:
802:
800:
716:
714:
623:
589:
542:
519:
501:
480:
457:
439:
421:
409:
401:
352:
328:
305:
287:
237:
219:
196:
178:
150:
138:
130:
4737:. These are, respectively, the standard
2600:Other names used for these matrices are
5939:Matrix representation of conic sections
5016:Problems and Theorems in Linear Algebra
4984:"Some Theorems on Commutative Matrices"
4879:
4877:
4875:
4871:
2487:is the product of its diagonal entries
1768:using the previously solved values for
393:, and analogously a matrix of the form
4170:{\displaystyle p(A_{1},\ldots ,A_{k})}
2640:All finite unitriangular matrices are
1533:{\displaystyle \ell _{1,1}x_{1}=b_{1}}
4638:Borel subgroups and Borel subalgebras
3607:containing all of the eigenvalues of
3444:{\displaystyle (e_{1},\ldots ,e_{n})}
2620:triangular matrix is not the same as
7:
1813:{\displaystyle x_{1},\dots ,x_{k-1}}
4762:
4591:
4561:
4551:
4538:
4514:
4483:
4086:{\displaystyle A_{1},\ldots ,A_{k}}
3888:{\displaystyle A_{1},\ldots ,A_{k}}
3706:{\displaystyle A_{1},\ldots ,A_{k}}
2271:{\displaystyle p_{A}(x)=\det(xI-A)}
1734:{\displaystyle x_{1},\dots ,x_{k}}
687:to triangular matrices are called
25:
4575:{\displaystyle {\mathfrak {n}}=.}
3611:(for example, any matrix over an
1484:Observe that the first equation (
5973:
4494:{\displaystyle {\mathfrak {n}}.}
3657:Simultaneous triangularisability
3592:All flags are conjugate (as the
938:
930:
905:
897:
5841:Used in science and engineering
4770:{\displaystyle {\mathfrak {b}}}
4699:of this group and the group of
4599:{\displaystyle {\mathfrak {n}}}
4522:{\displaystyle {\mathfrak {b}}}
4444:. It is often referred to as a
4351:Algebras of triangular matrices
3717:simultaneously triangularisable
3368:{\displaystyle i,j=1,\ldots ,k}
3046:{\displaystyle i,j=1,\ldots ,k}
2727:A block triangular matrix is a
103:and an upper triangular matrix
5084:Explicitly constrained entries
4566:
4546:
4303:
4277:
4225:
4199:
4164:
4138:
4135:
4103:
4004:
3972:
3949:
3917:
3797:
3765:
3438:
3406:
2563:
2541:
2535:
2516:
2513:
2494:
2474:
2459:
2395:
2373:
2367:
2348:
2345:
2326:
2320:
2314:
2265:
2250:
2241:
2235:
1820:. The resulting formulas are:
1675:. Continuing in this way, the
886:A matrix equation in the form
1:
5858:Fundamental (computer vision)
4922:Bernstein, Dennis S. (2009).
4660:, which is a subgroup of the
4025:of the polynomial algebra in
3663:Simultaneously diagonalizable
2203:times on the diagonal, where
1567:, and thus one can solve for
882:Forward and back substitution
99:of a lower triangular matrix
80:the main diagonal are zero.
27:Special kind of square matrix
4192:-commuting variables, where
52:. A square matrix is called
5624:Duplication and elimination
5423:eigenvalues or eigenvectors
4884:Axler, Sheldon Jay (1997).
4366:matrices, multiplied using
2711:Gauss transformation matrix
2412:that is, the unique degree
1695:-th equation only involves
6033:
5557:With specific applications
5186:Discrete Fourier Transform
4641:
4630:
4451:All these results hold if
4375:operations. They form the
3660:
3613:algebraically closed field
2720:
2682:
2648:Strictly triangular matrix
2480:{\displaystyle \det(xI-A)}
2195:In fact more is true: the
110:all its leading principal
36:
30:Not to be confused with a
29:
5967:
5848:Cabibbo–Kobayashi–Maskawa
5475:Satisfying conditions on
4887:Linear Algebra Done Right
3905:Hilbert's Nullstellensatz
2217:characteristic polynomial
1080:, then substituting that
1022:, and repeats through to
792:is lower triangular, and
6012:Numerical linear algebra
5013:Prasolov, V. V. (1994).
5000:10.1112/jlms/s1-26.3.221
4947:Herstein, I. N. (1975).
2679:Atomic triangular matrix
2671:as a consequence of the
1741:, and one can solve for
1121:, and repeating through
987:, then substitutes that
5206:Generalized permutation
4032:This is generalized by
3451:and the resulting flag
2717:Block triangular matrix
2673:Cayley-Hamilton theorem
1114:{\displaystyle x_{n-1}}
662:right triangular matrix
658:upper triangular matrix
387:lower triangular matrix
18:Upper triangular matrix
5980:Mathematics portal
4845:Cholesky decomposition
4819:Möbius transformations
4771:
4720:
4688:
4600:
4576:
4523:
4495:
4391:
4341:
4310:
4263:
4232:
4171:
4087:
4023:algebra representation
4011:
3956:
3889:
3843:
3807:
3749:
3748:{\displaystyle A_{i},}
3707:
3586:
3445:
3369:
3325:
3256:
3083:lower block triangular
3075:
3057:Lower block triangular
3047:
3003:
2934:
2761:upper block triangular
2753:
2735:Upper block triangular
2589:If the entries on the
2570:
2481:
2443:
2402:
2272:
2213:multiplicity as a root
2209:algebraic multiplicity
2103:
2046:
1814:
1762:
1735:
1689:
1669:
1642:
1615:
1588:
1561:
1534:
1475:
1142:
1115:
1088:equation to solve for
1074:
1043:
1016:
995:equation to solve for
981:
946:
913:
869:
783:
647:
391:left triangular matrix
376:
122:A matrix of the form
95:may be written as the
39:triangular matrix ring
4777:of the Lie algebra gl
4772:
4721:
4719:{\displaystyle \pm 1}
4689:
4687:{\displaystyle 2^{n}}
4601:
4577:
4524:
4496:
4472:nilpotent Lie algebra
4358:
4342:
4340:{\displaystyle A_{k}}
4311:
4264:
4262:{\displaystyle A_{i}}
4233:
4172:
4088:
4012:
3957:
3890:
3844:
3808:
3750:
3708:
3587:
3446:
3370:
3326:
3257:
3076:
3048:
3004:
2935:
2754:
2699:off-diagonal elements
2571:
2482:
2444:
2403:
2273:
2104:
2020:
1815:
1763:
1761:{\displaystyle x_{k}}
1736:
1690:
1670:
1668:{\displaystyle x_{1}}
1643:
1641:{\displaystyle x_{2}}
1616:
1614:{\displaystyle x_{1}}
1589:
1587:{\displaystyle x_{1}}
1562:
1560:{\displaystyle x_{1}}
1535:
1476:
1143:
1141:{\displaystyle x_{1}}
1116:
1075:
1073:{\displaystyle x_{n}}
1044:
1042:{\displaystyle x_{n}}
1017:
1015:{\displaystyle x_{2}}
982:
980:{\displaystyle x_{1}}
947:
914:
878:is upper triangular.
870:
784:
648:
377:
48:is a special kind of
4835:Gaussian elimination
4757:
4735:solvable Lie algebra
4707:
4671:
4662:general linear group
4586:
4533:
4509:
4501:This algebra is the
4478:
4442:solvable Lie algebra
4362:lower unitriangular
4324:
4274:
4246:
4196:
4181:for all polynomials
4097:
4051:
4038:solvable Lie algebra
3966:
3911:
3853:
3827:
3759:
3729:
3671:
3594:general linear group
3455:
3403:
3335:
3270:
3092:
3065:
3013:
2948:
2770:
2743:
2585:Unitriangular matrix
2491:
2453:
2442:{\displaystyle xI-A}
2424:
2301:
2222:
1827:
1772:
1745:
1699:
1679:
1652:
1625:
1598:
1571:
1544:
1488:
1177:
1159:The matrix equation
1155:Forward substitution
1125:
1092:
1057:
1026:
999:
964:
954:forward substitution
923:
890:
799:
713:
683:. Matrices that are
400:
129:
34:, a related concept.
5929:Linear independence
5176:Diagonally dominant
4744:of the Lie group GL
4618:functional analysis
4503:derived Lie algebra
4421:associative algebra
4042:abelian Lie algebra
3842:{\displaystyle A,B}
3643:Schur decomposition
3379:Triangularisability
2168:conjugate transpose
76:if all the entries
60:if all the entries
37:For the rings, see
5934:Matrix exponential
5924:Jordan normal form
5758:Fisher information
5629:Euclidean distance
5543:Totally unimodular
4860:Invariant subspace
4855:Tridiagonal matrix
4767:
4716:
4697:semidirect product
4684:
4596:
4572:
4519:
4491:
4392:
4386:and correspond to
4337:
4306:
4259:
4228:
4167:
4083:
4007:
3952:
3898:commuting matrices
3885:
3849:or more generally
3839:
3822:commuting matrices
3806:{\displaystyle K.}
3803:
3745:
3703:
3667:A set of matrices
3645:. This means that
3628:Jordan normal form
3582:
3441:
3365:
3321:
3252:
3246:
3071:
3043:
2999:
2930:
2924:
2749:
2566:
2477:
2439:
2398:
2268:
2099:
2097:
1810:
1758:
1731:
1685:
1665:
1638:
1611:
1584:
1557:
1530:
1471:
1469:
1138:
1111:
1070:
1039:
1012:
977:
942:
909:
865:
859:
779:
773:
643:
637:
372:
366:
85:numerical analysis
44:In mathematics, a
5999:
5998:
5991:Category:Matrices
5863:Fuzzy associative
5753:Doubly stochastic
5461:Positive-definite
5141:Block tridiagonal
4950:Topics in Algebra
4933:978-0-691-14039-1
4850:Hessenberg matrix
4750:and the standard
4701:diagonal matrices
4010:{\displaystyle K}
3955:{\displaystyle K}
3383:A matrix that is
3074:{\displaystyle A}
2752:{\displaystyle A}
2695:triangular matrix
2693:(upper or lower)
2667:of index at most
2656:(upper or lower)
2612:(upper or lower)
2608:, or very rarely
2604:(upper or lower)
2090:
1979:
1976:
1964:
1879:
1688:{\displaystyle k}
958:back substitution
93:invertible matrix
46:triangular matrix
16:(Redirected from
6024:
5986:List of matrices
5978:
5977:
5954:Row echelon form
5898:State transition
5827:Seidel adjacency
5709:Totally positive
5569:Alternating sign
5166:Complex Hadamard
5069:
5062:
5055:
5046:
5039:
5038:
5010:
5004:
5003:
4979:
4973:
4972:
4944:
4938:
4937:
4919:
4910:
4909:
4881:
4840:QR decomposition
4823:Heisenberg group
4776:
4774:
4773:
4768:
4766:
4765:
4752:Borel subalgebra
4725:
4723:
4722:
4717:
4693:
4691:
4690:
4685:
4683:
4682:
4648:Borel subalgebra
4605:
4603:
4602:
4597:
4595:
4594:
4581:
4579:
4578:
4573:
4565:
4564:
4555:
4554:
4542:
4541:
4528:
4526:
4525:
4520:
4518:
4517:
4500:
4498:
4497:
4492:
4487:
4486:
4457:lower triangular
4453:upper triangular
4446:Borel subalgebra
4439:
4346:
4344:
4343:
4338:
4336:
4335:
4315:
4313:
4312:
4309:{\displaystyle }
4307:
4302:
4301:
4289:
4288:
4268:
4266:
4265:
4260:
4258:
4257:
4242:; for commuting
4237:
4235:
4234:
4231:{\displaystyle }
4229:
4224:
4223:
4211:
4210:
4176:
4174:
4173:
4168:
4163:
4162:
4150:
4149:
4134:
4133:
4115:
4114:
4092:
4090:
4089:
4084:
4082:
4081:
4063:
4062:
4016:
4014:
4013:
4008:
4003:
4002:
3984:
3983:
3961:
3959:
3958:
3953:
3948:
3947:
3929:
3928:
3894:
3892:
3891:
3886:
3884:
3883:
3865:
3864:
3848:
3846:
3845:
3840:
3815:Borel subalgebra
3812:
3810:
3809:
3804:
3796:
3795:
3777:
3776:
3754:
3752:
3751:
3746:
3741:
3740:
3719:
3718:
3712:
3710:
3709:
3704:
3702:
3701:
3683:
3682:
3591:
3589:
3588:
3583:
3578:
3577:
3565:
3561:
3560:
3559:
3541:
3540:
3517:
3513:
3512:
3511:
3499:
3498:
3481:
3477:
3476:
3450:
3448:
3447:
3442:
3437:
3436:
3418:
3417:
3389:triangularizable
3374:
3372:
3371:
3366:
3330:
3328:
3327:
3322:
3320:
3319:
3318:
3317:
3305:
3304:
3294:
3285:
3284:
3261:
3259:
3258:
3253:
3251:
3250:
3243:
3242:
3223:
3222:
3208:
3207:
3159:
3158:
3147:
3146:
3118:
3117:
3080:
3078:
3077:
3072:
3052:
3050:
3049:
3044:
3008:
3006:
3005:
3000:
2998:
2997:
2996:
2995:
2983:
2982:
2972:
2963:
2962:
2939:
2937:
2936:
2931:
2929:
2928:
2921:
2920:
2867:
2866:
2847:
2846:
2828:
2827:
2808:
2807:
2796:
2795:
2758:
2756:
2755:
2750:
2703:Frobenius matrix
2685:Frobenius matrix
2575:
2573:
2572:
2567:
2562:
2561:
2534:
2533:
2512:
2511:
2486:
2484:
2483:
2478:
2448:
2446:
2445:
2440:
2407:
2405:
2404:
2399:
2394:
2393:
2366:
2365:
2344:
2343:
2313:
2312:
2277:
2275:
2274:
2269:
2234:
2233:
2108:
2106:
2105:
2100:
2098:
2091:
2089:
2088:
2073:
2072:
2071:
2062:
2061:
2045:
2034:
2016:
2015:
2005:
1996:
1995:
1977:
1974:
1972:
1965:
1963:
1962:
1947:
1946:
1945:
1936:
1935:
1917:
1916:
1906:
1897:
1896:
1880:
1878:
1877:
1862:
1861:
1852:
1843:
1842:
1819:
1817:
1816:
1811:
1809:
1808:
1784:
1783:
1767:
1765:
1764:
1759:
1757:
1756:
1740:
1738:
1737:
1732:
1730:
1729:
1711:
1710:
1694:
1692:
1691:
1686:
1674:
1672:
1671:
1666:
1664:
1663:
1647:
1645:
1644:
1639:
1637:
1636:
1620:
1618:
1617:
1612:
1610:
1609:
1593:
1591:
1590:
1585:
1583:
1582:
1566:
1564:
1563:
1558:
1556:
1555:
1540:) only involves
1539:
1537:
1536:
1531:
1529:
1528:
1516:
1515:
1506:
1505:
1480:
1478:
1477:
1472:
1470:
1466:
1465:
1449:
1448:
1439:
1438:
1406:
1405:
1396:
1395:
1373:
1372:
1363:
1362:
1339:
1338:
1337:
1331:
1325:
1316:
1315:
1300:
1299:
1298:
1297:
1295:
1294:
1285:
1284:
1262:
1261:
1252:
1251:
1232:
1231:
1216:
1215:
1214:
1213:
1212:
1211:
1209:
1208:
1199:
1198:
1147:
1145:
1144:
1139:
1137:
1136:
1120:
1118:
1117:
1112:
1110:
1109:
1079:
1077:
1076:
1071:
1069:
1068:
1053:first computing
1048:
1046:
1045:
1040:
1038:
1037:
1021:
1019:
1018:
1013:
1011:
1010:
986:
984:
983:
978:
976:
975:
951:
949:
948:
943:
941:
933:
918:
916:
915:
910:
908:
900:
874:
872:
871:
866:
864:
863:
788:
786:
785:
780:
778:
777:
689:triangularisable
652:
650:
649:
644:
642:
641:
634:
633:
617:
616:
615:
606:
605:
578:
577:
576:
558:
557:
553:
552:
530:
529:
512:
511:
495:
491:
490:
468:
467:
450:
449:
432:
431:
381:
379:
378:
373:
371:
370:
363:
362:
345:
344:
316:
315:
298:
297:
279:
256:
255:
248:
247:
230:
229:
211:
210:
209:
207:
206:
189:
188:
165:
164:
163:
161:
160:
89:LU decomposition
74:
73:
72:upper triangular
58:
57:
56:lower triangular
32:triangular array
21:
6032:
6031:
6027:
6026:
6025:
6023:
6022:
6021:
6002:
6001:
6000:
5995:
5972:
5963:
5912:
5836:
5782:
5718:
5552:
5470:
5416:
5355:
5156:Centrosymmetric
5079:
5073:
5043:
5042:
5027:
5012:
5011:
5007:
4981:
4980:
4976:
4961:
4946:
4945:
4941:
4934:
4921:
4920:
4913:
4898:
4883:
4882:
4873:
4868:
4831:
4807:
4792:Borel subgroups
4782:
4755:
4754:
4749:
4705:
4704:
4674:
4669:
4668:
4650:
4642:Main articles:
4640:
4635:
4611:Engel's theorem
4584:
4583:
4531:
4530:
4507:
4506:
4476:
4475:
4455:is replaced by
4435:
4384:
4373:
4353:
4327:
4322:
4321:
4293:
4280:
4272:
4271:
4249:
4244:
4243:
4215:
4202:
4194:
4193:
4154:
4141:
4125:
4106:
4095:
4094:
4073:
4054:
4049:
4048:
3994:
3975:
3964:
3963:
3939:
3920:
3909:
3908:
3875:
3856:
3851:
3850:
3825:
3824:
3787:
3768:
3757:
3756:
3732:
3727:
3726:
3716:
3715:
3713:are said to be
3693:
3674:
3669:
3668:
3665:
3659:
3569:
3551:
3532:
3531:
3527:
3503:
3490:
3489:
3485:
3468:
3464:
3453:
3452:
3428:
3409:
3401:
3400:
3381:
3333:
3332:
3309:
3296:
3289:
3273:
3268:
3267:
3245:
3244:
3231:
3229:
3224:
3211:
3209:
3196:
3193:
3192:
3187:
3182:
3177:
3171:
3170:
3165:
3160:
3150:
3148:
3138:
3135:
3134:
3129:
3124:
3119:
3109:
3102:
3090:
3089:
3063:
3062:
3059:
3011:
3010:
2987:
2974:
2967:
2951:
2946:
2945:
2923:
2922:
2909:
2907:
2902:
2897:
2891:
2890:
2885:
2880:
2875:
2869:
2868:
2855:
2853:
2848:
2838:
2836:
2830:
2829:
2816:
2814:
2809:
2799:
2797:
2787:
2780:
2768:
2767:
2741:
2740:
2737:
2725:
2719:
2687:
2681:
2650:
2587:
2582:
2550:
2525:
2503:
2489:
2488:
2451:
2450:
2422:
2421:
2382:
2357:
2335:
2304:
2299:
2298:
2225:
2220:
2219:
2211:, that is, its
2138:
2128:to construct a
2122:
2096:
2095:
2074:
2063:
2047:
2007:
2006:
1997:
1987:
1984:
1983:
1970:
1969:
1948:
1937:
1921:
1908:
1907:
1898:
1888:
1885:
1884:
1863:
1853:
1844:
1834:
1825:
1824:
1794:
1775:
1770:
1769:
1748:
1743:
1742:
1721:
1702:
1697:
1696:
1677:
1676:
1655:
1650:
1649:
1628:
1623:
1622:
1601:
1596:
1595:
1574:
1569:
1568:
1547:
1542:
1541:
1520:
1507:
1491:
1486:
1485:
1468:
1467:
1457:
1455:
1450:
1440:
1424:
1422:
1417:
1412:
1407:
1397:
1381:
1379:
1374:
1364:
1348:
1345:
1344:
1336:
1330:
1324:
1318:
1317:
1307:
1305:
1296:
1286:
1270:
1268:
1263:
1253:
1237:
1234:
1233:
1223:
1221:
1210:
1200:
1184:
1175:
1174:
1157:
1128:
1123:
1122:
1095:
1090:
1089:
1060:
1055:
1054:
1029:
1024:
1023:
1002:
997:
996:
967:
962:
961:
921:
920:
888:
887:
884:
858:
857:
852:
847:
841:
840:
835:
830:
824:
823:
818:
813:
803:
797:
796:
772:
771:
766:
761:
755:
754:
749:
744:
738:
737:
732:
727:
717:
711:
710:
704:
636:
635:
619:
614:
608:
607:
585:
583:
574:
573:
568:
563:
555:
554:
538:
536:
531:
515:
513:
497:
493:
492:
476:
474:
469:
453:
451:
435:
433:
417:
410:
398:
397:
365:
364:
348:
346:
324:
322:
317:
301:
299:
283:
280:
278:
273:
268:
263:
257:
254:
249:
233:
231:
215:
212:
208:
192:
190:
174:
171:
170:
162:
146:
139:
127:
126:
120:
71:
70:
55:
54:
42:
35:
28:
23:
22:
15:
12:
11:
5:
6030:
6028:
6020:
6019:
6014:
6004:
6003:
5997:
5996:
5994:
5993:
5988:
5983:
5968:
5965:
5964:
5962:
5961:
5956:
5951:
5946:
5944:Perfect matrix
5941:
5936:
5931:
5926:
5920:
5918:
5914:
5913:
5911:
5910:
5905:
5900:
5895:
5890:
5885:
5880:
5875:
5870:
5865:
5860:
5855:
5850:
5844:
5842:
5838:
5837:
5835:
5834:
5829:
5824:
5819:
5814:
5809:
5804:
5799:
5793:
5791:
5784:
5783:
5781:
5780:
5775:
5770:
5765:
5760:
5755:
5750:
5745:
5740:
5735:
5729:
5727:
5720:
5719:
5717:
5716:
5714:Transformation
5711:
5706:
5701:
5696:
5691:
5686:
5681:
5676:
5671:
5666:
5661:
5656:
5651:
5646:
5641:
5636:
5631:
5626:
5621:
5616:
5611:
5606:
5601:
5596:
5591:
5586:
5581:
5576:
5571:
5566:
5560:
5558:
5554:
5553:
5551:
5550:
5545:
5540:
5535:
5530:
5525:
5520:
5515:
5510:
5505:
5500:
5491:
5485:
5483:
5472:
5471:
5469:
5468:
5463:
5458:
5453:
5451:Diagonalizable
5448:
5443:
5438:
5433:
5427:
5425:
5421:Conditions on
5418:
5417:
5415:
5414:
5409:
5404:
5399:
5394:
5389:
5384:
5379:
5374:
5369:
5363:
5361:
5357:
5356:
5354:
5353:
5348:
5343:
5338:
5333:
5328:
5323:
5318:
5313:
5308:
5303:
5301:Skew-symmetric
5298:
5296:Skew-Hermitian
5293:
5288:
5283:
5278:
5273:
5268:
5263:
5258:
5253:
5248:
5243:
5238:
5233:
5228:
5223:
5218:
5213:
5208:
5203:
5198:
5193:
5188:
5183:
5178:
5173:
5168:
5163:
5158:
5153:
5148:
5143:
5138:
5133:
5131:Block-diagonal
5128:
5123:
5118:
5113:
5108:
5106:Anti-symmetric
5103:
5101:Anti-Hermitian
5098:
5093:
5087:
5085:
5081:
5080:
5074:
5072:
5071:
5064:
5057:
5049:
5041:
5040:
5025:
5005:
4994:(3): 221–228.
4974:
4959:
4939:
4932:
4911:
4896:
4870:
4869:
4867:
4864:
4863:
4862:
4857:
4852:
4847:
4842:
4837:
4830:
4827:
4815:additive group
4806:
4803:
4778:
4764:
4745:
4739:Borel subgroup
4715:
4712:
4681:
4677:
4644:Borel subgroup
4639:
4636:
4626:Hilbert spaces
4593:
4571:
4568:
4563:
4558:
4553:
4548:
4545:
4540:
4516:
4490:
4485:
4413:
4412:
4409:
4402:
4399:
4382:
4371:
4352:
4349:
4334:
4330:
4305:
4300:
4296:
4292:
4287:
4283:
4279:
4256:
4252:
4227:
4222:
4218:
4214:
4209:
4205:
4201:
4166:
4161:
4157:
4153:
4148:
4144:
4140:
4137:
4132:
4128:
4124:
4121:
4118:
4113:
4109:
4105:
4102:
4080:
4076:
4072:
4069:
4066:
4061:
4057:
4006:
4001:
3997:
3993:
3990:
3987:
3982:
3978:
3974:
3971:
3951:
3946:
3942:
3938:
3935:
3932:
3927:
3923:
3919:
3916:
3882:
3878:
3874:
3871:
3868:
3863:
3859:
3838:
3835:
3832:
3802:
3799:
3794:
3790:
3786:
3783:
3780:
3775:
3771:
3767:
3764:
3744:
3739:
3735:
3700:
3696:
3692:
3689:
3686:
3681:
3677:
3658:
3655:
3651:unitary matrix
3581:
3576:
3572:
3568:
3564:
3558:
3554:
3550:
3547:
3544:
3539:
3535:
3530:
3526:
3523:
3520:
3516:
3510:
3506:
3502:
3497:
3493:
3488:
3484:
3480:
3475:
3471:
3467:
3463:
3460:
3440:
3435:
3431:
3427:
3424:
3421:
3416:
3412:
3408:
3380:
3377:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3316:
3312:
3308:
3303:
3299:
3293:
3288:
3283:
3280:
3276:
3264:
3263:
3249:
3241:
3238:
3234:
3230:
3228:
3225:
3221:
3218:
3214:
3210:
3206:
3203:
3199:
3195:
3194:
3191:
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3172:
3169:
3166:
3164:
3161:
3157:
3153:
3149:
3145:
3141:
3137:
3136:
3133:
3130:
3128:
3125:
3123:
3120:
3116:
3112:
3108:
3107:
3105:
3100:
3097:
3070:
3058:
3055:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
2994:
2990:
2986:
2981:
2977:
2971:
2966:
2961:
2958:
2954:
2942:
2941:
2927:
2919:
2916:
2912:
2908:
2906:
2903:
2901:
2898:
2896:
2893:
2892:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2870:
2865:
2862:
2858:
2854:
2852:
2849:
2845:
2841:
2837:
2835:
2832:
2831:
2826:
2823:
2819:
2815:
2813:
2810:
2806:
2802:
2798:
2794:
2790:
2786:
2785:
2783:
2778:
2775:
2748:
2736:
2733:
2721:Main article:
2718:
2715:
2683:Main article:
2680:
2677:
2649:
2646:
2586:
2583:
2581:
2578:
2565:
2560:
2557:
2553:
2549:
2546:
2543:
2540:
2537:
2532:
2528:
2524:
2521:
2518:
2515:
2510:
2506:
2502:
2499:
2496:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2438:
2435:
2432:
2429:
2410:
2409:
2397:
2392:
2389:
2385:
2381:
2378:
2375:
2372:
2369:
2364:
2360:
2356:
2353:
2350:
2347:
2342:
2338:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2311:
2307:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2232:
2228:
2137:
2134:
2121:
2118:
2110:
2109:
2094:
2087:
2084:
2081:
2077:
2070:
2066:
2060:
2057:
2054:
2050:
2044:
2041:
2038:
2033:
2030:
2027:
2023:
2019:
2014:
2010:
2003:
2000:
1998:
1994:
1990:
1986:
1985:
1982:
1973:
1971:
1968:
1961:
1958:
1955:
1951:
1944:
1940:
1934:
1931:
1928:
1924:
1920:
1915:
1911:
1904:
1901:
1899:
1895:
1891:
1887:
1886:
1883:
1876:
1873:
1870:
1866:
1860:
1856:
1850:
1847:
1845:
1841:
1837:
1833:
1832:
1807:
1804:
1801:
1797:
1793:
1790:
1787:
1782:
1778:
1755:
1751:
1728:
1724:
1720:
1717:
1714:
1709:
1705:
1684:
1662:
1658:
1635:
1631:
1608:
1604:
1581:
1577:
1554:
1550:
1527:
1523:
1519:
1514:
1510:
1504:
1501:
1498:
1494:
1482:
1481:
1464:
1460:
1456:
1454:
1451:
1447:
1443:
1437:
1434:
1431:
1427:
1423:
1421:
1418:
1416:
1413:
1411:
1408:
1404:
1400:
1394:
1391:
1388:
1384:
1380:
1378:
1375:
1371:
1367:
1361:
1358:
1355:
1351:
1347:
1346:
1343:
1340:
1335:
1332:
1329:
1326:
1323:
1320:
1319:
1314:
1310:
1306:
1304:
1301:
1293:
1289:
1283:
1280:
1277:
1273:
1269:
1267:
1264:
1260:
1256:
1250:
1247:
1244:
1240:
1236:
1235:
1230:
1226:
1222:
1220:
1217:
1207:
1203:
1197:
1194:
1191:
1187:
1183:
1182:
1156:
1153:
1135:
1131:
1108:
1105:
1102:
1098:
1067:
1063:
1036:
1032:
1009:
1005:
974:
970:
940:
936:
932:
928:
907:
903:
899:
895:
883:
880:
876:
875:
862:
856:
853:
851:
848:
846:
843:
842:
839:
836:
834:
831:
829:
826:
825:
822:
819:
817:
814:
812:
809:
808:
806:
790:
789:
776:
770:
767:
765:
762:
760:
757:
756:
753:
750:
748:
745:
743:
740:
739:
736:
733:
731:
728:
726:
723:
722:
720:
703:
700:
654:
653:
640:
632:
629:
626:
622:
618:
613:
610:
609:
604:
601:
598:
595:
592:
588:
584:
582:
579:
575:
572:
569:
567:
564:
562:
559:
556:
551:
548:
545:
541:
537:
535:
532:
528:
525:
522:
518:
514:
510:
507:
504:
500:
496:
494:
489:
486:
483:
479:
475:
473:
470:
466:
463:
460:
456:
452:
448:
445:
442:
438:
434:
430:
427:
424:
420:
416:
415:
413:
408:
405:
383:
382:
369:
361:
358:
355:
351:
347:
343:
340:
337:
334:
331:
327:
323:
321:
318:
314:
311:
308:
304:
300:
296:
293:
290:
286:
282:
281:
277:
274:
272:
269:
267:
264:
262:
259:
258:
253:
250:
246:
243:
240:
236:
232:
228:
225:
222:
218:
214:
213:
205:
202:
199:
195:
191:
187:
184:
181:
177:
173:
172:
169:
166:
159:
156:
153:
149:
145:
144:
142:
137:
134:
119:
116:
114:are non-zero.
108:if and only if
91:algorithm, an
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6029:
6018:
6015:
6013:
6010:
6009:
6007:
5992:
5989:
5987:
5984:
5982:
5981:
5976:
5970:
5969:
5966:
5960:
5957:
5955:
5952:
5950:
5949:Pseudoinverse
5947:
5945:
5942:
5940:
5937:
5935:
5932:
5930:
5927:
5925:
5922:
5921:
5919:
5917:Related terms
5915:
5909:
5908:Z (chemistry)
5906:
5904:
5901:
5899:
5896:
5894:
5891:
5889:
5886:
5884:
5881:
5879:
5876:
5874:
5871:
5869:
5866:
5864:
5861:
5859:
5856:
5854:
5851:
5849:
5846:
5845:
5843:
5839:
5833:
5830:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5808:
5805:
5803:
5800:
5798:
5795:
5794:
5792:
5790:
5785:
5779:
5776:
5774:
5771:
5769:
5766:
5764:
5761:
5759:
5756:
5754:
5751:
5749:
5746:
5744:
5741:
5739:
5736:
5734:
5731:
5730:
5728:
5726:
5721:
5715:
5712:
5710:
5707:
5705:
5702:
5700:
5697:
5695:
5692:
5690:
5687:
5685:
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5665:
5662:
5660:
5657:
5655:
5652:
5650:
5647:
5645:
5642:
5640:
5637:
5635:
5632:
5630:
5627:
5625:
5622:
5620:
5617:
5615:
5612:
5610:
5607:
5605:
5602:
5600:
5597:
5595:
5592:
5590:
5587:
5585:
5582:
5580:
5577:
5575:
5572:
5570:
5567:
5565:
5562:
5561:
5559:
5555:
5549:
5546:
5544:
5541:
5539:
5536:
5534:
5531:
5529:
5526:
5524:
5521:
5519:
5516:
5514:
5511:
5509:
5506:
5504:
5501:
5499:
5495:
5492:
5490:
5487:
5486:
5484:
5482:
5478:
5473:
5467:
5464:
5462:
5459:
5457:
5454:
5452:
5449:
5447:
5444:
5442:
5439:
5437:
5434:
5432:
5429:
5428:
5426:
5424:
5419:
5413:
5410:
5408:
5405:
5403:
5400:
5398:
5395:
5393:
5390:
5388:
5385:
5383:
5380:
5378:
5375:
5373:
5370:
5368:
5365:
5364:
5362:
5358:
5352:
5349:
5347:
5344:
5342:
5339:
5337:
5334:
5332:
5329:
5327:
5324:
5322:
5319:
5317:
5314:
5312:
5309:
5307:
5304:
5302:
5299:
5297:
5294:
5292:
5289:
5287:
5284:
5282:
5279:
5277:
5274:
5272:
5269:
5267:
5266:Pentadiagonal
5264:
5262:
5259:
5257:
5254:
5252:
5249:
5247:
5244:
5242:
5239:
5237:
5234:
5232:
5229:
5227:
5224:
5222:
5219:
5217:
5214:
5212:
5209:
5207:
5204:
5202:
5199:
5197:
5194:
5192:
5189:
5187:
5184:
5182:
5179:
5177:
5174:
5172:
5169:
5167:
5164:
5162:
5159:
5157:
5154:
5152:
5149:
5147:
5144:
5142:
5139:
5137:
5134:
5132:
5129:
5127:
5124:
5122:
5119:
5117:
5114:
5112:
5109:
5107:
5104:
5102:
5099:
5097:
5096:Anti-diagonal
5094:
5092:
5089:
5088:
5086:
5082:
5077:
5070:
5065:
5063:
5058:
5056:
5051:
5050:
5047:
5036:
5032:
5028:
5026:9780821802366
5022:
5018:
5017:
5009:
5006:
5001:
4997:
4993:
4989:
4985:
4978:
4975:
4970:
4966:
4962:
4960:0-471-01090-1
4956:
4952:
4951:
4943:
4940:
4935:
4929:
4925:
4918:
4916:
4912:
4907:
4903:
4899:
4897:0-387-22595-1
4893:
4889:
4888:
4880:
4878:
4876:
4872:
4865:
4861:
4858:
4856:
4853:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4832:
4828:
4826:
4824:
4820:
4816:
4812:
4804:
4802:
4800:
4795:
4793:
4789:
4788:standard flag
4784:
4781:
4753:
4748:
4743:
4740:
4736:
4732:
4727:
4713:
4710:
4702:
4698:
4679:
4675:
4665:
4663:
4659:
4655:
4649:
4645:
4637:
4634:
4629:
4627:
4623:
4622:nest algebras
4620:which yields
4619:
4614:
4612:
4607:
4582:In addition,
4569:
4556:
4543:
4504:
4488:
4473:
4468:
4466:
4461:
4458:
4454:
4449:
4447:
4443:
4438:
4434:
4431:given by the
4430:
4426:
4422:
4418:
4410:
4407:
4403:
4400:
4397:
4396:
4395:
4389:
4385:
4378:
4374:
4370:
4365:
4361:
4357:
4350:
4348:
4332:
4328:
4319:
4298:
4294:
4290:
4285:
4281:
4254:
4250:
4241:
4220:
4216:
4212:
4207:
4203:
4191:
4188:
4184:
4180:
4159:
4155:
4151:
4146:
4142:
4130:
4126:
4122:
4119:
4116:
4111:
4107:
4100:
4078:
4074:
4070:
4067:
4064:
4059:
4055:
4045:
4043:
4039:
4035:
4034:Lie's theorem
4030:
4028:
4024:
4020:
3999:
3995:
3991:
3988:
3985:
3980:
3976:
3969:
3944:
3940:
3936:
3933:
3930:
3925:
3921:
3914:
3906:
3901:
3899:
3880:
3876:
3872:
3869:
3866:
3861:
3857:
3836:
3833:
3830:
3823:
3818:
3816:
3800:
3792:
3788:
3784:
3781:
3778:
3773:
3769:
3762:
3742:
3737:
3733:
3724:
3720:
3698:
3694:
3690:
3687:
3684:
3679:
3675:
3664:
3656:
3654:
3652:
3648:
3644:
3640:
3635:
3633:
3629:
3624:
3622:
3618:
3614:
3610:
3606:
3602:
3597:
3595:
3579:
3574:
3570:
3566:
3562:
3556:
3552:
3548:
3545:
3542:
3537:
3533:
3528:
3524:
3521:
3518:
3514:
3508:
3504:
3500:
3495:
3491:
3486:
3482:
3478:
3473:
3469:
3465:
3461:
3458:
3433:
3429:
3425:
3422:
3419:
3414:
3410:
3398:
3397:standard flag
3394:
3390:
3386:
3378:
3376:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3314:
3310:
3306:
3301:
3297:
3286:
3281:
3278:
3274:
3247:
3239:
3236:
3232:
3226:
3219:
3216:
3212:
3204:
3201:
3197:
3189:
3184:
3179:
3174:
3167:
3162:
3155:
3151:
3143:
3139:
3131:
3126:
3121:
3114:
3110:
3103:
3098:
3095:
3088:
3087:
3086:
3084:
3068:
3056:
3054:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
2992:
2988:
2984:
2979:
2975:
2964:
2959:
2956:
2952:
2925:
2917:
2914:
2910:
2904:
2899:
2894:
2887:
2882:
2877:
2872:
2863:
2860:
2856:
2850:
2843:
2839:
2833:
2824:
2821:
2817:
2811:
2804:
2800:
2792:
2788:
2781:
2776:
2773:
2766:
2765:
2764:
2762:
2746:
2734:
2732:
2730:
2724:
2716:
2714:
2712:
2708:
2704:
2700:
2696:
2692:
2686:
2678:
2676:
2674:
2670:
2666:
2661:
2659:
2655:
2647:
2645:
2643:
2638:
2636:
2632:
2628:
2627:
2623:
2619:
2616:. However, a
2615:
2611:
2607:
2603:
2598:
2596:
2595:unitriangular
2592:
2591:main diagonal
2584:
2580:Special forms
2579:
2577:
2558:
2555:
2551:
2547:
2544:
2538:
2530:
2526:
2522:
2519:
2508:
2504:
2500:
2497:
2471:
2468:
2465:
2462:
2436:
2433:
2430:
2427:
2419:
2415:
2390:
2387:
2383:
2379:
2376:
2370:
2362:
2358:
2354:
2351:
2340:
2336:
2332:
2329:
2323:
2317:
2309:
2305:
2297:
2296:
2295:
2293:
2289:
2285:
2281:
2262:
2259:
2256:
2253:
2244:
2238:
2230:
2226:
2218:
2214:
2210:
2206:
2202:
2198:
2193:
2191:
2187:
2182:
2180:
2176:
2173:
2169:
2165:
2161:
2157:
2154:
2150:
2145:
2143:
2135:
2133:
2131:
2127:
2126:bootstrapping
2119:
2117:
2115:
2092:
2085:
2082:
2079:
2075:
2068:
2064:
2058:
2055:
2052:
2048:
2042:
2039:
2036:
2031:
2028:
2025:
2021:
2017:
2012:
2008:
2001:
1999:
1992:
1988:
1980:
1966:
1959:
1956:
1953:
1949:
1942:
1938:
1932:
1929:
1926:
1922:
1918:
1913:
1909:
1902:
1900:
1893:
1889:
1881:
1874:
1871:
1868:
1864:
1858:
1854:
1848:
1846:
1839:
1835:
1823:
1822:
1821:
1805:
1802:
1799:
1795:
1791:
1788:
1785:
1780:
1776:
1753:
1749:
1726:
1722:
1718:
1715:
1712:
1707:
1703:
1682:
1660:
1656:
1633:
1629:
1606:
1602:
1579:
1575:
1552:
1548:
1525:
1521:
1517:
1512:
1508:
1502:
1499:
1496:
1492:
1462:
1458:
1452:
1445:
1441:
1435:
1432:
1429:
1425:
1419:
1414:
1409:
1402:
1398:
1392:
1389:
1386:
1382:
1376:
1369:
1365:
1359:
1356:
1353:
1349:
1341:
1333:
1327:
1321:
1312:
1308:
1302:
1291:
1287:
1281:
1278:
1275:
1271:
1265:
1258:
1254:
1248:
1245:
1242:
1238:
1228:
1224:
1218:
1205:
1201:
1195:
1192:
1189:
1185:
1173:
1172:
1171:
1169:
1165:
1162:
1154:
1152:
1149:
1133:
1129:
1106:
1103:
1100:
1096:
1087:
1083:
1065:
1061:
1052:
1034:
1030:
1007:
1003:
994:
990:
972:
968:
959:
955:
934:
926:
901:
893:
881:
879:
860:
854:
849:
844:
837:
832:
827:
820:
815:
810:
804:
795:
794:
793:
774:
768:
763:
758:
751:
746:
741:
734:
729:
724:
718:
709:
708:
707:
701:
699:
697:
692:
690:
686:
682:
677:
675:
671:
667:
663:
659:
656:is called an
638:
630:
627:
624:
620:
611:
602:
599:
596:
593:
590:
586:
580:
570:
565:
560:
549:
546:
543:
539:
533:
526:
523:
520:
516:
508:
505:
502:
498:
487:
484:
481:
477:
471:
464:
461:
458:
454:
446:
443:
440:
436:
428:
425:
422:
418:
411:
406:
403:
396:
395:
394:
392:
388:
367:
359:
356:
353:
349:
341:
338:
335:
332:
329:
325:
319:
312:
309:
306:
302:
294:
291:
288:
284:
275:
270:
265:
260:
251:
244:
241:
238:
234:
226:
223:
220:
216:
203:
200:
197:
193:
185:
182:
179:
175:
167:
157:
154:
151:
147:
140:
135:
132:
125:
124:
123:
117:
115:
113:
109:
106:
102:
98:
94:
90:
86:
81:
79:
75:
67:
66:main diagonal
63:
59:
51:
50:square matrix
47:
40:
33:
19:
5971:
5903:Substitution
5789:graph theory
5330:
5286:Quaternionic
5276:Persymmetric
5015:
5008:
4991:
4987:
4977:
4949:
4942:
4923:
4886:
4808:
4798:
4796:
4785:
4779:
4746:
4741:
4728:
4666:
4651:
4633:Affine group
4615:
4609:In fact, by
4608:
4469:
4462:
4456:
4452:
4450:
4436:
4414:
4393:
4377:Cayley table
4368:
4317:
4189:
4186:
4182:
4046:
4031:
4026:
4018:
3902:
3819:
3722:
3714:
3666:
3646:
3638:
3636:
3631:
3625:
3620:
3616:
3608:
3600:
3598:
3388:
3382:
3265:
3082:
3060:
2943:
2760:
2738:
2729:block matrix
2726:
2723:Block matrix
2710:
2707:Gauss matrix
2706:
2702:
2694:
2690:
2688:
2668:
2662:
2657:
2653:
2651:
2639:
2630:
2624:
2621:
2617:
2613:
2609:
2605:
2601:
2599:
2594:
2588:
2417:
2413:
2411:
2291:
2287:
2283:
2279:
2204:
2200:
2194:
2183:
2178:
2174:
2171:
2163:
2159:
2155:
2152:
2146:
2139:
2123:
2120:Applications
2113:
2111:
1483:
1167:
1163:
1160:
1158:
1150:
1085:
1081:
1050:
992:
988:
957:
953:
885:
877:
791:
705:
693:
688:
678:
673:
669:
665:
661:
657:
655:
390:
386:
385:is called a
384:
121:
104:
100:
82:
77:
69:
61:
53:
45:
43:
5878:Hamiltonian
5802:Biadjacency
5738:Correlation
5654:Householder
5604:Commutation
5341:Vandermonde
5336:Tridiagonal
5271:Permutation
5261:Nonnegative
5246:Matrix unit
5126:Bisymmetric
4731:Lie algebra
4656:, indeed a
4429:Lie bracket
4425:Lie algebra
4029:variables.
2635:matrix norm
2626:unit matrix
2294:is exactly
2197:eigenvalues
2186:determinant
2130:yield curve
706:The matrix
118:Description
6006:Categories
5778:Transition
5773:Stochastic
5743:Covariance
5725:statistics
5704:Symplectic
5699:Similarity
5528:Unimodular
5523:Orthogonal
5508:Involutory
5503:Invertible
5498:Projection
5494:Idempotent
5436:Convergent
5331:Triangular
5281:Polynomial
5226:Hessenberg
5196:Equivalent
5191:Elementary
5171:Copositive
5161:Conference
5121:Bidiagonal
4866:References
4811:isomorphic
4631:See also:
4474:, denoted
4433:commutator
4417:subalgebra
4240:commutator
3661:See also:
2658:triangular
2614:triangular
2606:triangular
2151:(meaning
2136:Properties
1051:backwards,
5959:Wronskian
5883:Irregular
5873:Gell-Mann
5822:Laplacian
5817:Incidence
5797:Adjacency
5768:Precision
5733:Centering
5639:Generator
5609:Confusion
5594:Circulant
5574:Augmented
5533:Unipotent
5513:Nilpotent
5489:Congruent
5466:Stieltjes
5441:Defective
5431:Companion
5402:Redheffer
5321:Symmetric
5316:Sylvester
5291:Signature
5221:Hermitian
5201:Frobenius
5111:Arrowhead
5091:Alternant
4711:±
4658:Lie group
4465:Lie group
4179:nilpotent
4120:…
4068:…
3989:…
3934:…
3870:…
3782:…
3688:…
3546:…
3522:⋯
3423:…
3357:…
3307:×
3287:∈
3227:⋯
3190:⋮
3185:⋱
3180:⋮
3175:⋮
3163:⋯
3127:⋯
3061:A matrix
3035:…
2985:×
2965:∈
2905:⋯
2888:⋮
2883:⋱
2878:⋮
2873:⋮
2851:⋯
2812:⋯
2739:A matrix
2665:nilpotent
2642:unipotent
2548:−
2539:⋯
2523:−
2501:−
2469:−
2434:−
2380:−
2371:⋯
2355:−
2333:−
2260:−
2190:permanent
2142:transpose
2076:ℓ
2049:ℓ
2040:−
2022:∑
2018:−
1981:⋮
1950:ℓ
1923:ℓ
1919:−
1865:ℓ
1803:−
1789:…
1716:…
1493:ℓ
1426:ℓ
1415:⋯
1383:ℓ
1350:ℓ
1342:⋮
1334:⋱
1328:⋮
1322:⋮
1272:ℓ
1239:ℓ
1186:ℓ
1104:−
1084:into the
991:into the
696:trapezoid
594:−
581:⋱
571:⋮
566:⋱
561:⋱
534:…
472:…
350:ℓ
339:−
326:ℓ
320:…
303:ℓ
285:ℓ
276:⋱
271:⋱
266:⋮
261:⋮
252:⋱
235:ℓ
217:ℓ
194:ℓ
176:ℓ
148:ℓ
87:. By the
6017:Matrices
5787:Used in
5723:Used in
5684:Rotation
5659:Jacobian
5619:Distance
5599:Cofactor
5584:Carleman
5564:Adjugate
5548:Weighing
5481:inverses
5477:products
5446:Definite
5377:Identity
5367:Exchange
5360:Constant
5326:Toeplitz
5211:Hadamard
5181:Diagonal
5035:30076024
4906:54850562
4829:See also
4805:Examples
4364:Toeplitz
4318:strictly
3755:denoted
3563:⟩
3529:⟨
3515:⟩
3487:⟨
3479:⟩
3466:⟨
3331:for all
3009:for all
2654:strictly
2629:, and a
2162:, where
1086:previous
702:Examples
681:diagonal
5888:Overlap
5853:Density
5812:Edmonds
5689:Seifert
5649:Hessian
5614:Coxeter
5538:Unitary
5456:Hurwitz
5387:Of ones
5372:Hilbert
5306:Skyline
5251:Metzler
5241:Logical
5236:Integer
5146:Boolean
5078:classes
4969:3307396
4813:to the
4437:ab − ba
4419:of the
4406:inverse
4238:is the
3603:over a
3385:similar
2709:, or a
2290:matrix
2215:of the
2207:is its
2166:is the
989:forward
685:similar
97:product
5807:Degree
5748:Design
5679:Random
5669:Payoff
5664:Moment
5589:Cartan
5579:Bézout
5518:Normal
5392:Pascal
5382:Lehmer
5311:Sparse
5231:Hollow
5216:Hankel
5151:Cauchy
5076:Matrix
5033:
5023:
4967:
4957:
4930:
4904:
4894:
4360:Binary
3641:has a
3266:where
2944:where
2691:atomic
2631:normed
2610:normed
2149:normal
1978:
1975:
112:minors
5868:Gamma
5832:Tutte
5694:Shear
5407:Shift
5397:Pauli
5346:Walsh
5256:Moore
5136:Block
4703:with
4654:group
3962:over
3605:field
78:below
62:above
5674:Pick
5644:Gram
5412:Zero
5116:Band
5031:OCLC
5021:ISBN
4965:OCLC
4955:ISBN
4928:ISBN
4902:OCLC
4892:ISBN
4729:The
4646:and
4404:The
3525:<
3519:<
3483:<
3462:<
3393:flag
2705:, a
2618:unit
2602:unit
2188:and
2184:The
2177:and
2140:The
1621:and
1082:back
993:next
64:the
5763:Hat
5496:or
5479:or
4996:doi
4799:not
4624:on
4505:of
4379:of
4316:is
4190:non
4185:in
4177:is
3085:if
3081:is
2763:if
2759:is
2689:An
2622:the
2457:det
2278:of
2248:det
919:or
672:or
660:or
389:or
6008::
5029:.
4992:26
4990:.
4986:.
4963:.
4914:^
4900:.
4874:^
4825:.
4783:.
4628:.
4467:.
3817:.
3723:P.
3375:.
3156:22
3144:21
3115:11
3053:.
2844:22
2805:12
2793:11
2713:.
2675:.
2660:.
2644:.
2637:.
2597:.
2576:.
2531:22
2509:11
2363:22
2341:11
2181:.
2179:AA
2160:AA
2158:=
2132:.
1166:=
1148:.
769:69
747:96
698:.
691:.
676:.
5893:S
5351:Z
5068:e
5061:t
5054:v
5037:.
5002:.
4998::
4971:.
4936:.
4908:.
4780:n
4763:b
4747:n
4742:B
4714:1
4680:n
4676:2
4592:n
4570:.
4567:]
4562:b
4557:,
4552:b
4547:[
4544:=
4539:n
4515:b
4489:.
4484:n
4390:.
4383:4
4381:Z
4372:2
4369:F
4333:k
4329:A
4304:]
4299:j
4295:A
4291:,
4286:i
4282:A
4278:[
4255:i
4251:A
4226:]
4221:j
4217:A
4213:,
4208:i
4204:A
4200:[
4187:k
4183:p
4165:]
4160:j
4156:A
4152:,
4147:i
4143:A
4139:[
4136:)
4131:k
4127:A
4123:,
4117:,
4112:1
4108:A
4104:(
4101:p
4079:k
4075:A
4071:,
4065:,
4060:1
4056:A
4027:k
4019:k
4005:]
4000:k
3996:x
3992:,
3986:,
3981:1
3977:x
3973:[
3970:K
3950:]
3945:k
3941:A
3937:,
3931:,
3926:1
3922:A
3918:[
3915:K
3881:k
3877:A
3873:,
3867:,
3862:1
3858:A
3837:B
3834:,
3831:A
3801:.
3798:]
3793:k
3789:A
3785:,
3779:,
3774:1
3770:A
3766:[
3763:K
3743:,
3738:i
3734:A
3699:k
3695:A
3691:,
3685:,
3680:1
3676:A
3647:A
3639:A
3632:A
3621:A
3617:A
3609:A
3601:A
3580:.
3575:n
3571:K
3567:=
3557:n
3553:e
3549:,
3543:,
3538:1
3534:e
3509:2
3505:e
3501:,
3496:1
3492:e
3474:1
3470:e
3459:0
3439:)
3434:n
3430:e
3426:,
3420:,
3415:1
3411:e
3407:(
3363:k
3360:,
3354:,
3351:1
3348:=
3345:j
3342:,
3339:i
3315:j
3311:n
3302:i
3298:n
3292:F
3282:j
3279:i
3275:A
3262:,
3248:]
3240:k
3237:k
3233:A
3220:2
3217:k
3213:A
3205:1
3202:k
3198:A
3168:0
3152:A
3140:A
3132:0
3122:0
3111:A
3104:[
3099:=
3096:A
3069:A
3041:k
3038:,
3032:,
3029:1
3026:=
3023:j
3020:,
3017:i
2993:j
2989:n
2980:i
2976:n
2970:F
2960:j
2957:i
2953:A
2940:,
2926:]
2918:k
2915:k
2911:A
2900:0
2895:0
2864:k
2861:2
2857:A
2840:A
2834:0
2825:k
2822:1
2818:A
2801:A
2789:A
2782:[
2777:=
2774:A
2747:A
2669:n
2564:)
2559:n
2556:n
2552:a
2545:x
2542:(
2536:)
2527:a
2520:x
2517:(
2514:)
2505:a
2498:x
2495:(
2475:)
2472:A
2466:I
2463:x
2460:(
2437:A
2431:I
2428:x
2418:A
2414:n
2408:,
2396:)
2391:n
2388:n
2384:a
2377:x
2374:(
2368:)
2359:a
2352:x
2349:(
2346:)
2337:a
2330:x
2327:(
2324:=
2321:)
2318:x
2315:(
2310:A
2306:p
2292:A
2288:n
2286:×
2284:n
2280:A
2266:)
2263:A
2257:I
2254:x
2251:(
2245:=
2242:)
2239:x
2236:(
2231:A
2227:p
2205:k
2201:k
2175:A
2172:A
2164:A
2156:A
2153:A
2114:U
2093:.
2086:m
2083:,
2080:m
2069:i
2065:x
2059:i
2056:,
2053:m
2043:1
2037:m
2032:1
2029:=
2026:i
2013:m
2009:b
2002:=
1993:m
1989:x
1967:,
1960:2
1957:,
1954:2
1943:1
1939:x
1933:1
1930:,
1927:2
1914:2
1910:b
1903:=
1894:2
1890:x
1882:,
1875:1
1872:,
1869:1
1859:1
1855:b
1849:=
1840:1
1836:x
1806:1
1800:k
1796:x
1792:,
1786:,
1781:1
1777:x
1754:k
1750:x
1727:k
1723:x
1719:,
1713:,
1708:1
1704:x
1683:k
1661:1
1657:x
1634:2
1630:x
1607:1
1603:x
1580:1
1576:x
1553:1
1549:x
1526:1
1522:b
1518:=
1513:1
1509:x
1503:1
1500:,
1497:1
1463:m
1459:b
1453:=
1446:m
1442:x
1436:m
1433:,
1430:m
1420:+
1410:+
1403:2
1399:x
1393:2
1390:,
1387:m
1377:+
1370:1
1366:x
1360:1
1357:,
1354:m
1313:2
1309:b
1303:=
1292:2
1288:x
1282:2
1279:,
1276:2
1266:+
1259:1
1255:x
1249:1
1246:,
1243:2
1229:1
1225:b
1219:=
1206:1
1202:x
1196:1
1193:,
1190:1
1168:b
1164:x
1161:L
1134:1
1130:x
1107:1
1101:n
1097:x
1066:n
1062:x
1035:n
1031:x
1008:2
1004:x
973:1
969:x
939:b
935:=
931:x
927:U
906:b
902:=
898:x
894:L
861:]
855:1
850:0
845:0
838:9
833:6
828:0
821:1
816:4
811:1
805:[
775:]
764:9
759:4
752:0
742:2
735:0
730:0
725:1
719:[
674:R
670:U
666:L
639:]
631:n
628:,
625:n
621:u
612:0
603:n
600:,
597:1
591:n
587:u
550:n
547:,
544:2
540:u
527:3
524:,
521:2
517:u
509:2
506:,
503:2
499:u
488:n
485:,
482:1
478:u
465:3
462:,
459:1
455:u
447:2
444:,
441:1
437:u
429:1
426:,
423:1
419:u
412:[
407:=
404:U
368:]
360:n
357:,
354:n
342:1
336:n
333:,
330:n
313:2
310:,
307:n
295:1
292:,
289:n
245:2
242:,
239:3
227:1
224:,
221:3
204:2
201:,
198:2
186:1
183:,
180:2
168:0
158:1
155:,
152:1
141:[
136:=
133:L
105:U
101:L
41:.
20:)
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