47:
7524:
437:
3258:
1569:
3446:
4403:
3042:
5197:
5340:
1420:
3298:
1848:
3995:
724:
5062:
732:
linear isometries—rotations, reflections, and their combinations—produce orthogonal matrices. The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the
4420:
takes advantage of many of the properties of orthogonal matrices for numerical linear algebra, and they arise naturally. For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonal change of bases; both take the form of orthogonal matrices. Having determinant ±1
5208:
4810:
5439:
2592:, and any orthogonal matrix can be produced by taking a rotation matrix and possibly negating all of its columns. This follows from the property of determinants that negating a column negates the determinant, and thus negating an odd (but not even) number of columns negates the determinant.
3253:{\displaystyle P^{\mathrm {T} }QP={\begin{bmatrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{bmatrix}}\ (n{\text{ even}}),\ P^{\mathrm {T} }QP={\begin{bmatrix}R_{1}&&&\\&\ddots &&\\&&R_{k}&\\&&&1\end{bmatrix}}\ (n{\text{ odd}}).}
2730:
3930:
2341:
1106:
1690:
2011:
601:
554:
1564:{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\text{ (rotation), }}\qquad {\begin{bmatrix}\cos \theta &\sin \theta \\\sin \theta &-\cos \theta \\\end{bmatrix}}{\text{ (reflection)}}}
925:
1320:
3441:{\displaystyle P^{\mathrm {T} }QP={\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}
4692:
440:
Visual understanding of multiplication by the transpose of a matrix. If A is an orthogonal matrix and B is its transpose, the ij-th element of the product AA will vanish if i≠j, because the i-th row of A is orthogonal to the j-th row of
5349:
4398:{\displaystyle \exp(\Omega )={\begin{bmatrix}1-2s^{2}+2x^{2}s^{2}&2xys^{2}-2zsc&2xzs^{2}+2ysc\\2xys^{2}+2zsc&1-2s^{2}+2y^{2}s^{2}&2yzs^{2}-2xsc\\2xzs^{2}-2ysc&2yzs^{2}+2xsc&1-2s^{2}+2z^{2}s^{2}\end{bmatrix}}.}
2639:
3819:
5192:{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.8125&0.0625\\3.4375&2.6875\end{bmatrix}}\rightarrow \cdots \rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}
5732:
4451:
Likewise, algorithms using
Householder and Givens matrices typically use specialized methods of multiplication and storage. For example, a Givens rotation affects only two rows of a matrix it multiplies, changing a full
2203:
2403:
6176:
There is no standard terminology for these matrices. They are variously called "semi-orthogonal matrices", "orthonormal matrices", "orthogonal matrices", and sometimes simply "matrices with orthonormal rows/columns".
3631:
5335:{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.41421&-1.06066\\1.06066&1.41421\end{bmatrix}}\rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}
5622:
2607:
orthogonal matrices with bottom right entry equal to 1. The remainder of the last column (and last row) must be zeros, and the product of any two such matrices has the same form. The rest of the matrix is an
983:
843:
1927:
1656:
1187:
2172:. It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy
2937:
479:
994:
3709:
186:
1197:
5040:
invariant under an orthogonal change of basis, such as the spectral norm or the
Frobenius norm.) For a near-orthogonal matrix, rapid convergence to the orthogonal factor can be achieved by a "
5964:
4466:. When uses of these reflections and rotations introduce zeros in a matrix, the space vacated is enough to store sufficient data to reproduce the transform, and to do so robustly. (Following
766:
of a molecule is a subgroup of O(3). Because floating point versions of orthogonal matrices have advantageous properties, they are key to many algorithms in numerical linear algebra, such as
5818:
1871:
Rotations become more complicated in higher dimensions; they can no longer be completely characterized by one angle, and may affect more than one planar subspace. It is common to describe a
5877:
2074:
acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a single subdiagonal entry. Any rotation matrix of size
1192:
267:
1843:{\displaystyle {\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}}{\text{ and }}{\begin{bmatrix}0&-1&0\\1&0&0\\0&0&-1\end{bmatrix}}}
2548:, with the projection map choosing or according to the determinant. Orthogonal matrices with determinant −1 do not include the identity, and so do not form a subgroup but only a
850:
3766:
3557:
2580:. In practical terms, a comparable statement is that any orthogonal matrix can be produced by taking a rotation matrix and possibly negating one of its columns, as we saw with
4429:
is 1 (which is the minimum), so errors are not magnified when multiplying with an orthogonal matrix. Many algorithms use orthogonal matrices like
Householder reflections and
719:{\displaystyle {\mathbf {v} }^{\mathrm {T} }{\mathbf {v} }=(Q{\mathbf {v} })^{\mathrm {T} }(Q{\mathbf {v} })={\mathbf {v} }^{\mathrm {T} }Q^{\mathrm {T} }Q{\mathbf {v} }.}
7182:
5626:
This may be combined with the
Babylonian method for extracting the square root of a matrix to give a recurrence which converges to an orthogonal matrix quadratically:
2738:
can reduce any orthogonal matrix to this constrained form, a series of such reflections can bring any orthogonal matrix to the identity; thus an orthogonal group is a
4444:(where permutations do the pivoting). However, they rarely appear explicitly as matrices; their special form allows more efficient representation, such as a list of
5629:
5016:
rotation matrix which has been computed as the composition of numerous twists and turns. Floating point does not match the mathematical ideal of real numbers, so
2348:
2345:
The converse is not true; having a determinant of ±1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample.
5520:. Construct a Householder reflection from the vector, then apply it to the smaller matrix (embedded in the larger size with a 1 at the bottom right corner).
4433:
for this reason. It is also helpful that, not only is an orthogonal matrix invertible, but its inverse is available essentially free, by exchanging indices.
3562:
5559:
4805:{\displaystyle R={\begin{bmatrix}\cdot &\cdot &\cdot \\0&\cdot &\cdot \\0&0&\cdot \\0&0&0\\0&0&0\end{bmatrix}}.}
7396:
4625:
5434:{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}0.393919&-0.919145\\0.919145&0.393919\end{bmatrix}}}
1605:
1136:
6615:
457:, and for matrices of complex numbers that leads instead to the unitary requirement. Orthogonal matrices preserve the dot product, so, for vectors
7487:
5460:, which essentially requires that the distribution not change if multiplied by any freely chosen orthogonal matrix. Orthogonalizing matrices with
2725:{\displaystyle {\begin{bmatrix}&&&0\\&\mathrm {O} (n)&&\vdots \\&&&0\\0&\cdots &0&1\end{bmatrix}}}
3925:{\displaystyle \Omega ={\begin{bmatrix}0&-z\theta &y\theta \\z\theta &0&-x\theta \\-y\theta &x\theta &0\end{bmatrix}}.}
2850:
6358:
3655:
132:
7406:
7172:
428:, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.
3021:
More broadly, the effect of any orthogonal matrix separates into independent actions on orthogonal two-dimensional subspaces. That is, if
2438:
The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. In fact, the set of all
5973:
A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not
932:
6265:
5881:
4815:
2336:{\displaystyle 1=\det(I)=\det \left(Q^{\mathrm {T} }Q\right)=\det \left(Q^{\mathrm {T} }\right)\det(Q)={\bigl (}\det(Q){\bigr )}^{2}.}
2053:
suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size
795:
5766:
5822:
5453:
90:
68:
5491:
later generalized as the "subgroup algorithm" (in which form it works just as well for permutations and rotations). To generate an
123:
5036:
orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular. (Closeness can be measured by any
1101:{\displaystyle {\begin{bmatrix}0&0&0&1\\0&0&1&0\\1&0&0&0\\0&1&0&0\end{bmatrix}}}
7207:
2411:, being +1 or −1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows.
224:
6754:
4555:
6581:
5537:
6451:
3524:
6971:
6608:
5541:
4522:
3473:
2423:
2006:{\displaystyle Q=I-2{\frac {{\mathbf {v} }{\mathbf {v} }^{\mathrm {T} }}{{\mathbf {v} }^{\mathrm {T} }{\mathbf {v} }}}.}
1879:, but this only works in three dimensions. Above three dimensions two or more angles are needed, each associated with a
1683:
Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for
7046:
6576:
2552:; it is also (separately) connected. Thus each orthogonal group falls into two pieces; and because the projection map
2839:
rotation matrix. Since the planes are fixed, each rotation has only one degree of freedom, its angle. By induction,
7202:
6724:
5344:
Gram-Schmidt yields an inferior solution, shown by a
Frobenius distance of 8.28659 instead of the minimum 8.12404.
4628:, as might occur with repeated measurements of a physical phenomenon to compensate for experimental errors. Write
1894:
The most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. Any
7306:
7177:
7091:
2408:
774:
549:{\displaystyle {\mathbf {u} }\cdot {\mathbf {v} }=\left(Q{\mathbf {u} }\right)\cdot \left(Q{\mathbf {v} }\right)}
6588:
5761:
Using a first-order approximation of the inverse and the same initialization results in the modified iteration:
5021:
4910:
are superfluous in the product, which is thus already in lower-triangular upper-triangular factored form, as in
1886:
However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.
7411:
7301:
7009:
6689:
5461:
2589:
2519:
729:
374:
61:
55:
6498:(1980), "The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators",
449:. Although we consider only real matrices here, the definition can be used for matrices with entries from any
920:{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}
7446:
7375:
7257:
7117:
6714:
6601:
1126:
matrices and , which we can interpret as the identity and a reflection of the real line across the origin.
6242:
5989:(except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a
5464:
uniformly distributed random entries does not result in uniformly distributed orthogonal matrices, but the
7316:
6899:
6704:
6417:
6076:
5986:
4915:
1915:
382:
370:
72:
5059:
For example, consider a non-orthogonal matrix for which the simple averaging algorithm takes seven steps
5028:
the columns, but it is not the most reliable, nor the most efficient, nor the most invariant method. The
3735:
3284:
rotation matrices, and with the remaining entries zero. Exceptionally, a rotation block may be diagonal,
7262:
6999:
6849:
6844:
6679:
6654:
6649:
4453:
3718:
2415:
7523:
6571:
2950:
Permutation matrices are simpler still; they form, not a Lie group, but only a finite group, the order
1687:
matrices and larger the non-rotational matrices can be more complicated than reflections. For example,
6226:
7456:
6814:
6644:
6624:
6553:
6507:
4911:
4483:
4437:
2015:
Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of
1854:
1315:{\displaystyle {\begin{aligned}1&=p^{2}+t^{2},\\1&=q^{2}+u^{2},\\0&=pq+tu.\end{aligned}}}
762:, which—with its subgroups—is widely used in mathematics and the physical sciences. For example, the
6422:
2200:
of any orthogonal matrix is +1 or −1. This follows from basic facts about determinants, as follows:
7477:
7451:
7029:
6834:
6824:
6495:
6456:
6346:
6207:
5540:. There are several different ways to get the unique solution, the simplest of which is taking the
5472:
5029:
4593:
2515:
2449:
747:
450:
436:
399:
321:
2989:. The even permutations produce the subgroup of permutation matrices of determinant +1, the order
7528:
7482:
7472:
7426:
7421:
7350:
7286:
7152:
6889:
6884:
6819:
6809:
6674:
6543:
6534:
Mezzadri, Francesco (2006), "How to generate random matrices from the classical compact groups",
6484:
6443:
6314:
6289:; Shahshahani, Mehrdad (1987), "The subgroup algorithm for generating uniform random variables",
6246:
5553:
5449:
4417:
3722:
2735:
2565:
1599:
5041:
736:
Orthogonal matrices are important for a number of reasons, both theoretical and practical. The
7560:
7539:
7326:
7321:
7311:
7291:
7252:
7247:
7076:
7056:
7051:
7042:
7037:
6984:
6879:
6829:
6774:
6744:
6739:
6719:
6709:
6669:
6523:
6476:
6435:
6394:
6354:
6306:
5025:
4951:
4422:
3010:
2414:
Stronger than the determinant restriction is the fact that an orthogonal matrix can always be
2150:
1880:
1860:
1664:
378:
317:
218:
2742:. The last column can be fixed to any unit vector, and each choice gives a different copy of
7534:
7502:
7431:
7370:
7365:
7345:
7281:
7187:
7157:
7142:
7127:
7122:
7061:
7014:
6989:
6979:
6950:
6869:
6864:
6839:
6769:
6749:
6659:
6639:
6515:
6468:
6427:
6386:
6298:
6230:
6195:
6064:
5751:
5465:
4559:
4494:
4426:
3729:
2739:
2485:
2113:
matrices, three such rotations suffice; and by fixing the sequence we can thus describe all
1668:
767:
751:
425:
410:
6262:
2128:
has the same form as a Givens rotation, but is used to zero both off-diagonal entries of a
7232:
7167:
7147:
7132:
7112:
7096:
6994:
6925:
6915:
6874:
6759:
6729:
6405:
6371:
6367:
6269:
5974:
4430:
3030:
2959:
2822:. A single rotation can produce a zero in the first row of the last column, and series of
2804:
2503:
2500:
2190:
2154:
2125:
2071:
1672:
1671:(equal to its transpose) as well as orthogonal. The product of two rotation matrices is a
571:
474:
366:
207:
3295:
reflection diagonalizes to a +1 and −1, any orthogonal matrix can be brought to the form
6557:
6511:
2117:
rotation matrices (though not uniquely) in terms of the three angles used, often called
7492:
7436:
7416:
7401:
7360:
7237:
7197:
7162:
7086:
7025:
7004:
6945:
6935:
6920:
6854:
6799:
6789:
6784:
6694:
6286:
5008:
The case of a square invertible matrix also holds interest. Suppose, for example, that
3477:
3469:
2553:
2530:
2427:
2419:
2146:
1876:
445:
An orthogonal matrix is the real specialization of a unitary matrix, and thus always a
297:
120:
104:
35:
31:
4436:
Permutations are essential to the success of many algorithms, including the workhorse
7554:
7497:
7355:
7296:
7227:
7217:
7212:
7137:
7066:
6940:
6930:
6859:
6779:
6764:
6699:
6488:
6342:
6318:
4969:
4441:
2453:
446:
358:
329:
116:
6447:
6408:; Schreiber, Robert (July 1990), "Fast polar decomposition of an arbitrary matrix",
3725:
of any skew-symmetric matrix is an orthogonal matrix (in fact, special orthogonal).
789:
Below are a few examples of small orthogonal matrices and possible interpretations.
17:
7380:
7337:
7242:
6955:
6894:
6804:
6684:
5998:
5990:
5457:
2767:
2118:
7222:
7192:
6960:
6794:
6664:
5037:
3714:
2197:
2163:
763:
560:
is an orthogonal matrix. To see the inner product connection, consider a vector
454:
350:
346:
7273:
6734:
6302:
6272:, Nicholas J. Higham, Mathematics of Computation, Volume 46, Number 174, 1986.
6057:
5456:
random orthogonal matrices. In this context, "uniform" is defined in terms of
1395:
is the identity), and the second as a reflection across a line at an angle of
354:
6527:
6480:
6439:
6398:
6310:
5727:{\displaystyle Q_{n+1}=2M\left(Q_{n}^{-1}M+M^{\mathrm {T} }Q_{n}\right)^{-1}}
4950:
In the case of a linear system which is underdetermined, or an otherwise non-
7507:
7081:
6018:
2951:
2456:
195:
2398:{\displaystyle {\begin{bmatrix}2&0\\0&{\frac {1}{2}}\end{bmatrix}}}
5548:
and replacing the singular values with ones. Another method expresses the
3934:
The exponential of this is the orthogonal matrix for rotation around axis
7441:
2166:, which is the case if and only if its rows form an orthonormal basis of
414:
362:
6548:
5056:
has published an accelerated method with a convenient convergence test.
4947:, but also for allowing solution without magnifying numerical problems.
1675:, and the product of two reflection matrices is also a rotation matrix.
6472:
6191:
5452:
and exploration of high-dimensional data spaces, require generation of
3626:{\displaystyle {\dot {Q}}^{\mathrm {T} }Q+Q^{\mathrm {T} }{\dot {Q}}=0}
1324:
In consideration of the first equation, without loss of generality let
4986:
merely replaces each non-zero diagonal entry with its reciprocal. Set
574:. Written with respect to an orthonormal basis, the squared length of
27:
Real square matrix whose columns and rows are orthogonal unit vectors
6519:
6431:
6390:
4474:
store a rotation angle, which is both expensive and badly behaved.)
3468:
give conjugate pairs of eigenvalues lying on the unit circle in the
3025:
is special orthogonal then one can always find an orthogonal matrix
424:
consisting of orthogonal matrices with determinant +1 is called the
5617:{\displaystyle Q=M\left(M^{\mathrm {T} }M\right)^{-{\frac {1}{2}}}}
3488:
rotation, the eigenvector associated with +1 is the rotation axis.
1904:
permutation matrix can be constructed as a product of no more than
2829:
rotations will zero all but the last row of the last column of an
2549:
6593:
978:{\displaystyle {\begin{bmatrix}1&0\\0&-1\\\end{bmatrix}}}
838:{\displaystyle {\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}}
733:
same dimension, and these have no orthogonal matrix equivalent.
6597:
5052:), repeatedly averaging the matrix with its inverse transpose.
3810:
being a unit vector, the correct skew-symmetric matrix form of
3484:
is odd, there is at least one real eigenvalue, +1 or −1; for a
1651:{\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}.}
1182:{\displaystyle {\begin{bmatrix}p&t\\q&u\end{bmatrix}},}
6250:
6040:
is simply connected and thus the universal covering group for
778:
40:
4954:, singular value decomposition (SVD) is equally useful. With
3732:
is a differential rotation, thus a vector in the Lie algebra
2807:
using an analogous procedure. The bundle structure persists:
6326:
3291:. Thus, negating one column if necessary, and noting that a
1602:, with a single 1 in each column and row (and otherwise 0):
6353:(3/e ed.), Baltimore: Johns Hopkins University Press,
4421:
and all eigenvalues of magnitude 1 is of great benefit for
6190:, matrices with orthonormal columns may be referred to as
6067:, which themselves can be built from orthogonal matrices.
2021:. This is a reflection in the hyperplane perpendicular to
6327:"An Optimum Iteration for the Matrix Polar Decomposition"
6291:
Probability in the
Engineering and Informational Sciences
5513:
one and a uniformly distributed unit vector of dimension
5032:
factors a matrix into a pair, one of which is the unique
4918:). Here orthogonality is important not only for reducing
4859:) are independent, the projection solution is found from
2932:{\displaystyle (n-1)+(n-2)+\cdots +1={\frac {n(n-1)}{2}}}
773:. As another example, with appropriate normalization the
3728:
For example, the three-dimensional object physics calls
2803:; and any special orthogonal matrix can be generated by
1189:
which orthogonality demands satisfy the three equations
213:
This leads to the equivalent characterization: a matrix
2499:
The orthogonal matrices whose determinant is +1 form a
5397:
5358:
5298:
5256:
5217:
5155:
5110:
5071:
4707:
4022:
3834:
3704:{\displaystyle {\dot {Q}}^{\mathrm {T} }=-{\dot {Q}}.}
3391:
3332:
3328:
3169:
3072:
2648:
2407:
With permutation matrices the determinant matches the
2357:
1776:
1699:
1614:
1580:
generates a reflection about the line at 45° given by
1498:
1429:
1145:
1003:
941:
859:
804:
181:{\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,}
6589:
6459:(1976), "The Economical Storage of Plane Rotations",
6372:"Computing the Polar Decomposition—with Applications"
5884:
5825:
5769:
5632:
5562:
5352:
5211:
5065:
4695:
4490:) involve orthogonal matrices, including especially:
3998:
3822:
3738:
3658:
3565:
3527:
3301:
3045:
2853:
2642:
2351:
2206:
1930:
1693:
1608:
1423:
1195:
1139:
997:
935:
853:
798:
781:
compression) is represented by an orthogonal matrix.
604:
482:
398:
orthogonal matrices, under multiplication, forms the
227:
135:
6410:
SIAM Journal on
Scientific and Statistical Computing
6379:
SIAM Journal on
Scientific and Statistical Computing
6048:. By far the most famous example of a spin group is
453:. However, orthogonal matrices arise naturally from
7465:
7389:
7335:
7271:
7105:
7023:
6969:
6908:
6632:
5488:
1384:. We can interpret the first case as a rotation by
5958:
5871:
5812:
5726:
5616:
5433:
5334:
5191:
4804:
4397:
3924:
3760:
3703:
3625:
3551:
3440:
3252:
2931:
2724:
2448:orthogonal matrices satisfies all the axioms of a
2397:
2335:
2005:
1842:
1650:
1563:
1314:
1181:
1100:
977:
919:
837:
718:
548:
353:of any orthogonal matrix is either +1 or −1. As a
261:
180:
4968:, a satisfactory solution uses the Moore-Penrose
2588:is odd, then the semidirect product is in fact a
5959:{\displaystyle Q_{n+1}=2Q_{n}+P_{n}N_{n}-3P_{n}}
5475:random entries does, as long as the diagonal of
5199:and which acceleration trims to two steps (with
2734:Since an elementary reflection in the form of a
2304:
2282:
2259:
2228:
2213:
5813:{\displaystyle N_{n}=Q_{n}^{\mathrm {T} }Q_{n}}
1573:The special case of the reflection matrix with
1108: (permutation of coordinate axes)
217:is orthogonal if its transpose is equal to its
5872:{\displaystyle P_{n}={\frac {1}{2}}Q_{n}N_{n}}
5487:replaced this with a more efficient idea that
3521:. Differentiating the orthogonality condition
6609:
6331:Electronic Transactions on Numerical Analysis
5528:The problem of finding the orthogonal matrix
5020:has gradually lost its true orthogonality. A
4487:
2319:
2299:
589:. If a linear transformation, in matrix form
8:
6536:Notices of the American Mathematical Society
6263:"Newton's Method for the Matrix Square Root"
6085:is not a square matrix, then the conditions
1667:, which implies that a reflection matrix is
2084:can be constructed as a product of at most
2063:can be constructed as a product of at most
2027:(negating any vector component parallel to
1660:The identity is also a permutation matrix.
7183:Fundamental (linear differential equation)
6616:
6602:
6594:
4847:to the subspace spanned by the columns of
3717:of an orthogonal matrix group consists of
3472:; so this decomposition confirms that all
927: (rotation about the origin)
6547:
6421:
6125:are orthonormal. This can only happen if
6063:The Pin and Spin groups are found within
5950:
5934:
5924:
5911:
5889:
5883:
5863:
5853:
5839:
5830:
5824:
5804:
5793:
5792:
5787:
5774:
5768:
5750:These iterations are stable provided the
5715:
5704:
5693:
5692:
5673:
5668:
5637:
5631:
5602:
5598:
5583:
5582:
5561:
5392:
5353:
5351:
5293:
5251:
5212:
5210:
5150:
5105:
5066:
5064:
4702:
4694:
4626:overdetermined system of linear equations
4378:
4368:
4352:
4316:
4280:
4242:
4221:
4211:
4195:
4159:
4121:
4085:
4064:
4054:
4038:
4017:
3997:
3829:
3821:
3740:
3739:
3737:
3687:
3686:
3673:
3672:
3661:
3660:
3657:
3606:
3605:
3598:
3597:
3580:
3579:
3568:
3567:
3564:
3533:
3532:
3526:
3390:
3366:
3339:
3331:
3323:
3307:
3306:
3300:
3239:
3205:
3176:
3164:
3148:
3147:
3129:
3106:
3079:
3067:
3051:
3050:
3044:
2902:
2852:
2662:
2643:
2641:
2377:
2352:
2350:
2324:
2318:
2317:
2298:
2297:
2271:
2270:
2241:
2240:
2205:
1991:
1990:
1983:
1982:
1976:
1975:
1965:
1964:
1958:
1957:
1950:
1949:
1946:
1929:
1771:
1766:
1694:
1692:
1609:
1607:
1556:
1493:
1487:
1424:
1422:
1267:
1254:
1227:
1214:
1196:
1194:
1140:
1138:
1122:The simplest orthogonal matrices are the
998:
996:
936:
934:
854:
852:
799:
797:
707:
706:
696:
695:
684:
683:
677:
676:
663:
662:
649:
648:
638:
637:
622:
621:
614:
613:
607:
606:
603:
535:
534:
512:
511:
494:
493:
484:
483:
481:
247:
233:
232:
226:
162:
161:
141:
140:
134:
91:Learn how and when to remove this message
30:For matrices with orthogonality over the
6243:"Finding the Nearest Orthonormal Matrix"
5480:
5053:
4556:Eigendecomposition of a symmetric matrix
3713:In Lie group terms, this means that the
845: (identity transformation)
435:
286:is necessarily invertible (with inverse
262:{\displaystyle Q^{\mathrm {T} }=Q^{-1},}
54:This article includes a list of general
7488:Matrix representation of conic sections
6219:
6149:(due to linear dependence). Similarly,
5484:
4467:
5977:to each other, even the +1 component,
5049:
5045:
1918:is constructed from a non-null vector
5552:explicitly but requires the use of a
5448:Some numerical applications, such as
361:of vectors, and therefore acts as an
357:, an orthogonal matrix preserves the
7:
4897:) and invertible, and also equal to
4841:, which is equivalent to projecting
3761:{\displaystyle {\mathfrak {so}}(3)}
3744:
3741:
3552:{\displaystyle Q^{\mathrm {T} }Q=I}
2426:, all of which must have (complex)
2145:A real square matrix is orthogonal
6500:SIAM Journal on Numerical Analysis
6108:are not equivalent. The condition
5794:
5694:
5584:
4008:
3823:
3674:
3599:
3581:
3534:
3308:
3149:
3052:
2663:
2272:
2242:
1984:
1966:
985: (reflection across
697:
685:
650:
615:
234:
163:
142:
60:it lacks sufficient corresponding
25:
5489:Diaconis & Shahshahani (1987)
4906:. But the lower rows of zeros in
3721:. Going the other direction, the
750:under matrix multiplication, the
598:, preserves vector lengths, then
7522:
6163:are orthonormal, which requires
5479:contains only positive entries (
4558:(decomposition according to the
3500:are differentiable functions of
2970:. By the same kind of argument,
2939:degrees of freedom, and so does
1992:
1977:
1959:
1951:
708:
678:
664:
639:
623:
608:
536:
513:
495:
485:
45:
7390:Used in science and engineering
4615:symmetric positive-semidefinite
4462:to a much more efficient order
2109:such rotations. In the case of
1875:rotation matrix in terms of an
1492:
6633:Explicitly constrained entries
6325:Dubrulle, Augustin A. (1999),
5389:
5290:
5248:
5147:
5141:
5102:
4425:. One implication is that the
4011:
4005:
3755:
3749:
3244:
3233:
3134:
3123:
2920:
2908:
2884:
2872:
2866:
2854:
2673:
2667:
2313:
2307:
2291:
2285:
2222:
2216:
669:
656:
645:
631:
1:
7407:Fundamental (computer vision)
6194:and they are elements of the
5538:Orthogonal Procrustes problem
5542:singular value decomposition
4523:Singular value decomposition
2634:(and of all higher groups).
2162:with the ordinary Euclidean
7173:Duplication and elimination
6972:eigenvalues or eigenvectors
6577:Encyclopedia of Mathematics
5503:orthogonal matrix, take an
746:orthogonal matrices form a
129:One way to express this is
119:whose columns and rows are
7577:
7106:With specific applications
6735:Discrete Fourier Transform
6227:"Paul's online math notes"
6074:
4851:. Assuming the columns of
3037:into block diagonal form:
1864:, respectively, about the
381:. In other words, it is a
29:
7516:
7397:Cabibbo–Kobayashi–Maskawa
7024:Satisfying conditions on
6303:10.1017/S0269964800000255
6121:says that the columns of
6017:has covering groups, the
5524:Nearest orthogonal matrix
4488:Golub & Van Loan 1996
2422:to exhibit a full set of
1858:through the origin and a
775:discrete cosine transform
4408:Numerical linear algebra
2618:orthogonal matrix; thus
2520:special orthogonal group
1590:and therefore exchanges
1113:Elementary constructions
426:special orthogonal group
6755:Generalized permutation
6056:, or the group of unit
6052:, which is nothing but
5532:nearest a given matrix
5203:= 0.353553, 0.565685).
4818:problem is to find the
3719:skew-symmetric matrices
3496:Suppose the entries of
2039:is a unit vector, then
1489: (rotation),
1133:matrices have the form
75:more precise citations.
7529:Mathematics portal
6159:says that the rows of
6077:Semi-orthogonal matrix
5960:
5873:
5814:
5728:
5618:
5435:
5336:
5193:
4916:Cholesky decomposition
4806:
4669:decomposition reduces
4482:A number of important
4399:
3926:
3762:
3705:
3627:
3553:
3442:
3254:
2933:
2805:Givens plane rotations
2726:
2399:
2337:
2007:
1916:Householder reflection
1844:
1652:
1565:
1316:
1183:
1102:
979:
921:
839:
720:
550:
442:
383:unitary transformation
263:
182:
6461:Numerische Mathematik
5961:
5874:
5815:
5729:
5619:
5454:uniformly distributed
5436:
5337:
5194:
4807:
4484:matrix decompositions
4400:
3927:
3763:
3706:
3628:
3554:
3443:
3255:
2934:
2770:over the unit sphere
2727:
2400:
2338:
2132:symmetric submatrix.
2008:
1845:
1653:
1566:
1317:
1184:
1103:
980:
922:
840:
721:
551:
439:
355:linear transformation
282:An orthogonal matrix
264:
183:
6347:Van Loan, Charles F.
6233:, 2008. Theorem 3(c)
6071:Rectangular matrices
5882:
5823:
5767:
5758:is less than three.
5630:
5560:
5473:normally distributed
5350:
5209:
5063:
5022:Gram–Schmidt process
4912:Gaussian elimination
4816:linear least squares
4693:
4673:to upper triangular
4438:Gaussian elimination
3996:
3820:
3736:
3656:
3563:
3525:
3299:
3043:
2851:
2640:
2349:
2204:
2149:its columns form an
1928:
1691:
1606:
1421:
1193:
1137:
995:
933:
851:
796:
602:
480:
225:
133:
18:Orthogonal transform
7478:Linear independence
6725:Diagonally dominant
6572:"Orthogonal matrix"
6558:2006math.ph...9050M
6512:1980SJNA...17..403S
6351:Matrix Computations
6208:Biorthogonal system
6192:orthogonal k-frames
5799:
5681:
5450:Monte Carlo methods
5030:polar decomposition
4594:Polar decomposition
3262:where the matrices
322:conjugate transpose
7483:Matrix exponential
7473:Jordan normal form
7307:Fisher information
7178:Euclidean distance
7092:Totally unimodular
6473:10.1007/BF01462266
6268:2011-09-29 at the
6247:Berthold K.P. Horn
5956:
5869:
5810:
5783:
5724:
5664:
5614:
5554:matrix square root
5536:is related to the
5431:
5425:
5383:
5332:
5326:
5284:
5242:
5189:
5183:
5135:
5096:
5044:" approach due to
4802:
4793:
4677:. For example, if
4418:Numerical analysis
4395:
4386:
3922:
3913:
3758:
3723:matrix exponential
3701:
3623:
3549:
3438:
3429:
3425:
3374:
3250:
3224:
3114:
2929:
2736:Householder matrix
2722:
2716:
2566:semidirect product
2529:of rotations. The
2395:
2389:
2333:
2067:such reflections.
2003:
1840:
1834:
1760:
1648:
1639:
1600:permutation matrix
1561:
1558: (reflection)
1550:
1481:
1312:
1310:
1179:
1170:
1098:
1092:
975:
969:
917:
911:
835:
829:
730:finite-dimensional
716:
570:-dimensional real
546:
473:-dimensional real
443:
275:is the inverse of
259:
178:
113:orthonormal matrix
7548:
7547:
7540:Category:Matrices
7412:Fuzzy associative
7302:Doubly stochastic
7010:Positive-definite
6690:Block tridiagonal
6360:978-0-8018-5414-9
6065:Clifford algebras
5847:
5610:
4952:invertible matrix
4423:numeric stability
3695:
3669:
3614:
3576:
3242:
3232:
3142:
3132:
3122:
3029:, a (rotational)
3011:alternating group
2979:is a subgroup of
2927:
2795:is a subgroup of
2626:is a subgroup of
2544:is isomorphic to
2385:
2151:orthonormal basis
2141:Matrix properties
1998:
1881:plane of rotation
1769:
1679:Higher dimensions
1559:
1490:
318:Hermitian adjoint
109:orthogonal matrix
101:
100:
93:
16:(Redirected from
7568:
7535:List of matrices
7527:
7526:
7503:Row echelon form
7447:State transition
7376:Seidel adjacency
7258:Totally positive
7118:Alternating sign
6715:Complex Hadamard
6618:
6611:
6604:
6595:
6585:
6560:
6551:
6530:
6491:
6450:
6425:
6406:Higham, Nicholas
6401:
6385:(4): 1160–1174,
6376:
6368:Higham, Nicholas
6363:
6338:
6321:
6273:
6260:
6254:
6240:
6234:
6231:Lamar University
6229:, Paul Dawkins,
6224:
6196:Stiefel manifold
6189:
6172:
6162:
6158:
6148:
6138:
6128:
6120:
6107:
6097:
6084:
6055:
6051:
6047:
6039:
6031:
6016:
6008:
5987:simply connected
5984:
5965:
5963:
5962:
5957:
5955:
5954:
5939:
5938:
5929:
5928:
5916:
5915:
5900:
5899:
5878:
5876:
5875:
5870:
5868:
5867:
5858:
5857:
5848:
5840:
5835:
5834:
5819:
5817:
5816:
5811:
5809:
5808:
5798:
5797:
5791:
5779:
5778:
5757:
5752:condition number
5746:
5733:
5731:
5730:
5725:
5723:
5722:
5714:
5710:
5709:
5708:
5699:
5698:
5697:
5680:
5672:
5648:
5647:
5623:
5621:
5620:
5615:
5613:
5612:
5611:
5603:
5597:
5593:
5589:
5588:
5587:
5551:
5547:
5535:
5531:
5519:
5512:
5502:
5478:
5468:
5440:
5438:
5437:
5432:
5430:
5429:
5388:
5387:
5341:
5339:
5338:
5333:
5331:
5330:
5289:
5288:
5247:
5246:
5202:
5198:
5196:
5195:
5190:
5188:
5187:
5140:
5139:
5101:
5100:
5019:
5015:
5011:
5004:
4991:
4985:
4981:
4967:
4957:
4946:
4937:
4909:
4905:
4896:
4886:
4877:
4858:
4854:
4850:
4846:
4840:
4838:
4823:
4811:
4809:
4808:
4803:
4798:
4797:
4688:
4684:
4680:
4676:
4672:
4668:
4664:
4654:
4644:
4640:
4614:
4610:
4606:
4589:
4585:
4581:
4577:
4560:spectral theorem
4551:
4547:
4543:
4539:
4519:upper triangular
4518:
4514:
4510:
4497:
4465:
4461:
4447:
4442:partial pivoting
4431:Givens rotations
4427:condition number
4404:
4402:
4401:
4396:
4391:
4390:
4383:
4382:
4373:
4372:
4357:
4356:
4321:
4320:
4285:
4284:
4247:
4246:
4226:
4225:
4216:
4215:
4200:
4199:
4164:
4163:
4126:
4125:
4090:
4089:
4069:
4068:
4059:
4058:
4043:
4042:
3991:
3990:
3988:
3987:
3984:
3981:
3967:
3966:
3964:
3963:
3960:
3957:
3943:
3939:
3931:
3929:
3928:
3923:
3918:
3917:
3815:
3809:
3790:
3771:
3767:
3765:
3764:
3759:
3748:
3747:
3730:angular velocity
3710:
3708:
3707:
3702:
3697:
3696:
3688:
3679:
3678:
3677:
3671:
3670:
3662:
3651:
3641:
3632:
3630:
3629:
3624:
3616:
3615:
3607:
3604:
3603:
3602:
3586:
3585:
3584:
3578:
3577:
3569:
3558:
3556:
3555:
3550:
3539:
3538:
3537:
3520:
3510:
3503:
3499:
3487:
3483:
3467:
3447:
3445:
3444:
3439:
3434:
3433:
3426:
3415:
3414:
3411:
3405:
3402:
3401:
3375:
3371:
3370:
3360:
3359:
3356:
3350:
3347:
3346:
3344:
3343:
3313:
3312:
3311:
3294:
3290:
3283:
3279:
3259:
3257:
3256:
3251:
3243:
3240:
3230:
3229:
3228:
3217:
3216:
3215:
3212:
3210:
3209:
3199:
3198:
3195:
3194:
3188:
3185:
3184:
3183:
3181:
3180:
3154:
3153:
3152:
3140:
3133:
3130:
3120:
3119:
3118:
3111:
3110:
3100:
3099:
3096:
3090:
3087:
3086:
3084:
3083:
3057:
3056:
3055:
3036:
3028:
3024:
3009:
3008:
3006:
3005:
3002:
2999:
2988:
2978:
2969:
2957:
2946:
2938:
2936:
2935:
2930:
2928:
2923:
2903:
2846:
2838:
2828:
2821:
2802:
2794:
2783:
2775:
2765:
2757:
2749:
2740:reflection group
2731:
2729:
2728:
2723:
2721:
2720:
2687:
2686:
2685:
2677:
2666:
2660:
2652:
2651:
2650:
2633:
2625:
2617:
2606:
2587:
2583:
2579:
2575:
2563:
2547:
2543:
2528:
2513:
2495:
2486:orthogonal group
2483:
2482:
2480:
2479:
2476:
2473:
2447:
2434:Group properties
2404:
2402:
2401:
2396:
2394:
2393:
2386:
2378:
2342:
2340:
2339:
2334:
2329:
2328:
2323:
2322:
2303:
2302:
2281:
2277:
2276:
2275:
2255:
2251:
2247:
2246:
2245:
2188:
2184:
2171:
2161:
2131:
2116:
2112:
2108:
2107:
2105:
2104:
2101:
2098:
2083:
2066:
2062:
2052:
2038:
2032:
2026:
2020:
2012:
2010:
2009:
2004:
1999:
1997:
1996:
1995:
1989:
1988:
1987:
1981:
1980:
1972:
1971:
1970:
1969:
1963:
1962:
1955:
1954:
1947:
1923:
1911:transpositions.
1910:
1903:
1874:
1867:
1849:
1847:
1846:
1841:
1839:
1838:
1770:
1767:
1765:
1764:
1686:
1663:A reflection is
1657:
1655:
1654:
1649:
1644:
1643:
1597:
1593:
1589:
1579:
1570:
1568:
1567:
1562:
1560:
1557:
1555:
1554:
1491:
1488:
1486:
1485:
1414:
1413:
1411:
1410:
1407:
1404:
1394:
1387:
1383:
1373:
1363:
1353:
1343:
1333:
1321:
1319:
1318:
1313:
1311:
1272:
1271:
1259:
1258:
1232:
1231:
1219:
1218:
1188:
1186:
1185:
1180:
1175:
1174:
1132:
1125:
1118:Lower dimensions
1107:
1105:
1104:
1099:
1097:
1096:
984:
982:
981:
976:
974:
973:
926:
924:
923:
918:
916:
915:
844:
842:
841:
836:
834:
833:
770:
761:
752:orthogonal group
745:
725:
723:
722:
717:
712:
711:
702:
701:
700:
690:
689:
688:
682:
681:
668:
667:
655:
654:
653:
643:
642:
627:
626:
620:
619:
618:
612:
611:
597:
588:
579:
569:
565:
559:
555:
553:
552:
547:
545:
541:
540:
539:
522:
518:
517:
516:
499:
498:
489:
488:
472:
468:
462:
423:
411:orthogonal group
408:
397:
344:
328:, and therefore
327:
315:
309:
295:
285:
278:
274:
268:
266:
265:
260:
255:
254:
239:
238:
237:
216:
205:
201:
193:
187:
185:
184:
179:
168:
167:
166:
147:
146:
145:
96:
89:
85:
82:
76:
71:this article by
62:inline citations
49:
48:
41:
21:
7576:
7575:
7571:
7570:
7569:
7567:
7566:
7565:
7551:
7550:
7549:
7544:
7521:
7512:
7461:
7385:
7331:
7267:
7101:
7019:
6965:
6904:
6705:Centrosymmetric
6628:
6622:
6570:
6567:
6549:math-ph/0609050
6533:
6520:10.1137/0717034
6494:
6455:
6432:10.1137/0911038
6423:10.1.1.230.4322
6404:
6391:10.1137/0907079
6374:
6366:
6361:
6341:
6324:
6287:Diaconis, Persi
6285:
6282:
6277:
6276:
6270:Wayback Machine
6261:
6257:
6241:
6237:
6225:
6221:
6216:
6204:
6181:
6164:
6160:
6150:
6140:
6130:
6126:
6109:
6099:
6086:
6082:
6079:
6073:
6053:
6049:
6041:
6033:
6026:
6010:
6002:
5978:
5971:
5946:
5930:
5920:
5907:
5885:
5880:
5879:
5859:
5849:
5826:
5821:
5820:
5800:
5770:
5765:
5764:
5755:
5741:
5735:
5700:
5688:
5663:
5659:
5658:
5633:
5628:
5627:
5578:
5577:
5573:
5572:
5558:
5557:
5549:
5545:
5533:
5529:
5526:
5514:
5504:
5492:
5476:
5471:of independent
5466:
5446:
5424:
5423:
5418:
5412:
5411:
5403:
5393:
5382:
5381:
5376:
5370:
5369:
5364:
5354:
5348:
5347:
5325:
5324:
5319:
5313:
5312:
5304:
5294:
5283:
5282:
5277:
5271:
5270:
5262:
5252:
5241:
5240:
5235:
5229:
5228:
5223:
5213:
5207:
5206:
5200:
5182:
5181:
5176:
5170:
5169:
5161:
5151:
5134:
5133:
5128:
5122:
5121:
5116:
5106:
5095:
5094:
5089:
5083:
5082:
5077:
5067:
5061:
5060:
5054:Dubrulle (1999)
5042:Newton's method
5017:
5013:
5009:
4993:
4987:
4983:
4973:
4959:
4955:
4939:
4919:
4907:
4898:
4888:
4879:
4860:
4856:
4852:
4848:
4842:
4827:
4825:
4824:that minimizes
4819:
4792:
4791:
4786:
4781:
4775:
4774:
4769:
4764:
4758:
4757:
4752:
4747:
4741:
4740:
4735:
4730:
4724:
4723:
4718:
4713:
4703:
4691:
4690:
4686:
4682:
4678:
4674:
4670:
4666:
4656:
4646:
4642:
4629:
4622:
4612:
4608:
4598:
4587:
4583:
4579:
4565:
4552:diagonal matrix
4549:
4545:
4541:
4527:
4516:
4512:
4502:
4495:
4480:
4463:
4457:
4445:
4415:
4410:
4385:
4384:
4374:
4364:
4348:
4337:
4312:
4301:
4276:
4264:
4263:
4238:
4227:
4217:
4207:
4191:
4180:
4155:
4143:
4142:
4117:
4106:
4081:
4070:
4060:
4050:
4034:
4018:
3994:
3993:
3985:
3982:
3977:
3976:
3974:
3969:
3961:
3958:
3953:
3952:
3950:
3945:
3941:
3935:
3912:
3911:
3906:
3898:
3886:
3885:
3874:
3869:
3860:
3859:
3851:
3840:
3830:
3818:
3817:
3811:
3792:
3773:
3769:
3734:
3733:
3659:
3654:
3653:
3652:) then implies
3643:
3636:
3593:
3566:
3561:
3560:
3528:
3523:
3522:
3512:
3505:
3501:
3497:
3494:
3485:
3481:
3466:
3457:
3451:
3428:
3427:
3424:
3423:
3412:
3410:
3403:
3400:
3388:
3382:
3381:
3376:
3373:
3372:
3362:
3357:
3355:
3348:
3345:
3335:
3324:
3302:
3297:
3296:
3292:
3285:
3281:
3278:
3269:
3263:
3223:
3222:
3213:
3211:
3201:
3196:
3193:
3186:
3182:
3172:
3165:
3143:
3113:
3112:
3102:
3097:
3095:
3088:
3085:
3075:
3068:
3046:
3041:
3040:
3034:
3031:change of basis
3026:
3022:
3019:
3003:
3000:
2994:
2993:
2991:
2990:
2987:
2980:
2977:
2971:
2968:
2962:
2960:symmetric group
2952:
2940:
2904:
2849:
2848:
2840:
2830:
2823:
2808:
2796:
2788:
2777:
2771:
2759:
2751:
2743:
2715:
2714:
2709:
2704:
2699:
2693:
2692:
2683:
2682:
2676:
2658:
2657:
2644:
2638:
2637:
2627:
2619:
2609:
2596:
2585:
2581:
2577:
2569:
2557:
2545:
2533:
2522:
2507:
2504:normal subgroup
2489:
2488:and denoted by
2477:
2474:
2464:
2463:
2461:
2460:
2439:
2436:
2420:complex numbers
2388:
2387:
2375:
2369:
2368:
2363:
2353:
2347:
2346:
2316:
2266:
2262:
2236:
2235:
2231:
2202:
2201:
2191:diagonal matrix
2186:
2173:
2167:
2157:
2155:Euclidean space
2143:
2138:
2129:
2126:Jacobi rotation
2114:
2110:
2102:
2099:
2089:
2088:
2086:
2085:
2075:
2072:Givens rotation
2064:
2054:
2040:
2034:
2028:
2022:
2016:
1974:
1973:
1956:
1948:
1926:
1925:
1919:
1905:
1895:
1892:
1872:
1865:
1833:
1832:
1824:
1819:
1813:
1812:
1807:
1802:
1796:
1795:
1790:
1782:
1772:
1768: and
1759:
1758:
1750:
1745:
1739:
1738:
1733:
1725:
1719:
1718:
1713:
1708:
1695:
1689:
1688:
1684:
1681:
1673:rotation matrix
1665:its own inverse
1638:
1637:
1632:
1626:
1625:
1620:
1610:
1604:
1603:
1595:
1591:
1581:
1574:
1549:
1548:
1534:
1522:
1521:
1510:
1494:
1480:
1479:
1468:
1456:
1455:
1441:
1425:
1419:
1418:
1408:
1405:
1400:
1399:
1397:
1396:
1389:
1385:
1375:
1365:
1355:
1345:
1335:
1325:
1309:
1308:
1283:
1277:
1276:
1263:
1250:
1243:
1237:
1236:
1223:
1210:
1203:
1191:
1190:
1169:
1168:
1163:
1157:
1156:
1151:
1141:
1135:
1134:
1130:
1123:
1120:
1115:
1091:
1090:
1085:
1080:
1075:
1069:
1068:
1063:
1058:
1053:
1047:
1046:
1041:
1036:
1031:
1025:
1024:
1019:
1014:
1009:
999:
993:
992:
968:
967:
959:
953:
952:
947:
937:
931:
930:
910:
909:
898:
886:
885:
871:
855:
849:
848:
828:
827:
822:
816:
815:
810:
800:
794:
793:
787:
768:
755:
737:
691:
675:
644:
605:
600:
599:
590:
581:
575:
572:Euclidean space
567:
561:
557:
530:
526:
507:
503:
478:
477:
475:Euclidean space
470:
464:
458:
434:
417:
409:, known as the
402:
389:
367:Euclidean space
333:
325:
311:
301:
287:
283:
276:
270:
243:
228:
223:
222:
214:
208:identity matrix
203:
199:
189:
157:
136:
131:
130:
97:
86:
80:
77:
67:Please help to
66:
50:
46:
39:
28:
23:
22:
15:
12:
11:
5:
7574:
7572:
7564:
7563:
7553:
7552:
7546:
7545:
7543:
7542:
7537:
7532:
7517:
7514:
7513:
7511:
7510:
7505:
7500:
7495:
7493:Perfect matrix
7490:
7485:
7480:
7475:
7469:
7467:
7463:
7462:
7460:
7459:
7454:
7449:
7444:
7439:
7434:
7429:
7424:
7419:
7414:
7409:
7404:
7399:
7393:
7391:
7387:
7386:
7384:
7383:
7378:
7373:
7368:
7363:
7358:
7353:
7348:
7342:
7340:
7333:
7332:
7330:
7329:
7324:
7319:
7314:
7309:
7304:
7299:
7294:
7289:
7284:
7278:
7276:
7269:
7268:
7266:
7265:
7263:Transformation
7260:
7255:
7250:
7245:
7240:
7235:
7230:
7225:
7220:
7215:
7210:
7205:
7200:
7195:
7190:
7185:
7180:
7175:
7170:
7165:
7160:
7155:
7150:
7145:
7140:
7135:
7130:
7125:
7120:
7115:
7109:
7107:
7103:
7102:
7100:
7099:
7094:
7089:
7084:
7079:
7074:
7069:
7064:
7059:
7054:
7049:
7040:
7034:
7032:
7021:
7020:
7018:
7017:
7012:
7007:
7002:
7000:Diagonalizable
6997:
6992:
6987:
6982:
6976:
6974:
6970:Conditions on
6967:
6966:
6964:
6963:
6958:
6953:
6948:
6943:
6938:
6933:
6928:
6923:
6918:
6912:
6910:
6906:
6905:
6903:
6902:
6897:
6892:
6887:
6882:
6877:
6872:
6867:
6862:
6857:
6852:
6850:Skew-symmetric
6847:
6845:Skew-Hermitian
6842:
6837:
6832:
6827:
6822:
6817:
6812:
6807:
6802:
6797:
6792:
6787:
6782:
6777:
6772:
6767:
6762:
6757:
6752:
6747:
6742:
6737:
6732:
6727:
6722:
6717:
6712:
6707:
6702:
6697:
6692:
6687:
6682:
6680:Block-diagonal
6677:
6672:
6667:
6662:
6657:
6655:Anti-symmetric
6652:
6650:Anti-Hermitian
6647:
6642:
6636:
6634:
6630:
6629:
6623:
6621:
6620:
6613:
6606:
6598:
6592:
6591:
6586:
6566:
6565:External links
6563:
6562:
6561:
6531:
6506:(3): 403–409,
6496:Stewart, G. W.
6492:
6467:(2): 137–138,
6457:Stewart, G. W.
6453:
6416:(4): 648–655,
6402:
6364:
6359:
6343:Golub, Gene H.
6339:
6322:
6281:
6278:
6275:
6274:
6255:
6235:
6218:
6217:
6215:
6212:
6211:
6210:
6203:
6200:
6075:Main article:
6072:
6069:
5991:covering group
5970:
5967:
5953:
5949:
5945:
5942:
5937:
5933:
5927:
5923:
5919:
5914:
5910:
5906:
5903:
5898:
5895:
5892:
5888:
5866:
5862:
5856:
5852:
5846:
5843:
5838:
5833:
5829:
5807:
5803:
5796:
5790:
5786:
5782:
5777:
5773:
5739:
5721:
5718:
5713:
5707:
5703:
5696:
5691:
5687:
5684:
5679:
5676:
5671:
5667:
5662:
5657:
5654:
5651:
5646:
5643:
5640:
5636:
5609:
5606:
5601:
5596:
5592:
5586:
5581:
5576:
5571:
5568:
5565:
5525:
5522:
5485:Stewart (1980)
5445:
5442:
5428:
5422:
5419:
5417:
5414:
5413:
5410:
5407:
5404:
5402:
5399:
5398:
5396:
5391:
5386:
5380:
5377:
5375:
5372:
5371:
5368:
5365:
5363:
5360:
5359:
5357:
5329:
5323:
5320:
5318:
5315:
5314:
5311:
5308:
5305:
5303:
5300:
5299:
5297:
5292:
5287:
5281:
5278:
5276:
5273:
5272:
5269:
5266:
5263:
5261:
5258:
5257:
5255:
5250:
5245:
5239:
5236:
5234:
5231:
5230:
5227:
5224:
5222:
5219:
5218:
5216:
5186:
5180:
5177:
5175:
5172:
5171:
5168:
5165:
5162:
5160:
5157:
5156:
5154:
5149:
5146:
5143:
5138:
5132:
5129:
5127:
5124:
5123:
5120:
5117:
5115:
5112:
5111:
5109:
5104:
5099:
5093:
5090:
5088:
5085:
5084:
5081:
5078:
5076:
5073:
5072:
5070:
4801:
4796:
4790:
4787:
4785:
4782:
4780:
4777:
4776:
4773:
4770:
4768:
4765:
4763:
4760:
4759:
4756:
4753:
4751:
4748:
4746:
4743:
4742:
4739:
4736:
4734:
4731:
4729:
4726:
4725:
4722:
4719:
4717:
4714:
4712:
4709:
4708:
4706:
4701:
4698:
4621:
4618:
4617:
4616:
4596:
4591:
4563:
4553:
4525:
4520:
4500:
4479:
4478:Decompositions
4476:
4468:Stewart (1976)
4454:multiplication
4414:
4411:
4409:
4406:
4394:
4389:
4381:
4377:
4371:
4367:
4363:
4360:
4355:
4351:
4347:
4344:
4341:
4338:
4336:
4333:
4330:
4327:
4324:
4319:
4315:
4311:
4308:
4305:
4302:
4300:
4297:
4294:
4291:
4288:
4283:
4279:
4275:
4272:
4269:
4266:
4265:
4262:
4259:
4256:
4253:
4250:
4245:
4241:
4237:
4234:
4231:
4228:
4224:
4220:
4214:
4210:
4206:
4203:
4198:
4194:
4190:
4187:
4184:
4181:
4179:
4176:
4173:
4170:
4167:
4162:
4158:
4154:
4151:
4148:
4145:
4144:
4141:
4138:
4135:
4132:
4129:
4124:
4120:
4116:
4113:
4110:
4107:
4105:
4102:
4099:
4096:
4093:
4088:
4084:
4080:
4077:
4074:
4071:
4067:
4063:
4057:
4053:
4049:
4046:
4041:
4037:
4033:
4030:
4027:
4024:
4023:
4021:
4016:
4013:
4010:
4007:
4004:
4001:
3921:
3916:
3910:
3907:
3905:
3902:
3899:
3897:
3894:
3891:
3888:
3887:
3884:
3881:
3878:
3875:
3873:
3870:
3868:
3865:
3862:
3861:
3858:
3855:
3852:
3850:
3847:
3844:
3841:
3839:
3836:
3835:
3833:
3828:
3825:
3757:
3754:
3751:
3746:
3743:
3700:
3694:
3691:
3685:
3682:
3676:
3668:
3665:
3635:Evaluation at
3622:
3619:
3613:
3610:
3601:
3596:
3592:
3589:
3583:
3575:
3572:
3548:
3545:
3542:
3536:
3531:
3493:
3490:
3478:absolute value
3462:
3455:
3437:
3432:
3422:
3419:
3416:
3413:
3409:
3406:
3404:
3399:
3396:
3393:
3392:
3389:
3387:
3384:
3383:
3380:
3377:
3369:
3365:
3361:
3358:
3354:
3351:
3349:
3342:
3338:
3334:
3333:
3330:
3329:
3327:
3322:
3319:
3316:
3310:
3305:
3274:
3267:
3249:
3246:
3238:
3235:
3227:
3221:
3218:
3214:
3208:
3204:
3200:
3197:
3192:
3189:
3187:
3179:
3175:
3171:
3170:
3168:
3163:
3160:
3157:
3151:
3146:
3139:
3136:
3128:
3125:
3117:
3109:
3105:
3101:
3098:
3094:
3091:
3089:
3082:
3078:
3074:
3073:
3071:
3066:
3063:
3060:
3054:
3049:
3033:, that brings
3018:
3017:Canonical form
3015:
2982:
2973:
2964:
2926:
2922:
2919:
2916:
2913:
2910:
2907:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2847:therefore has
2758:; in this way
2719:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2694:
2691:
2688:
2684:
2681:
2678:
2675:
2672:
2669:
2665:
2661:
2659:
2656:
2653:
2649:
2647:
2590:direct product
2531:quotient group
2501:path-connected
2435:
2432:
2392:
2384:
2381:
2376:
2374:
2371:
2370:
2367:
2364:
2362:
2359:
2358:
2356:
2332:
2327:
2321:
2315:
2312:
2309:
2306:
2301:
2296:
2293:
2290:
2287:
2284:
2280:
2274:
2269:
2265:
2261:
2258:
2254:
2250:
2244:
2239:
2234:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2147:if and only if
2142:
2139:
2137:
2134:
2002:
1994:
1986:
1979:
1968:
1961:
1953:
1945:
1942:
1939:
1936:
1933:
1891:
1888:
1877:axis and angle
1837:
1831:
1828:
1825:
1823:
1820:
1818:
1815:
1814:
1811:
1808:
1806:
1803:
1801:
1798:
1797:
1794:
1791:
1789:
1786:
1783:
1781:
1778:
1777:
1775:
1763:
1757:
1754:
1751:
1749:
1746:
1744:
1741:
1740:
1737:
1734:
1732:
1729:
1726:
1724:
1721:
1720:
1717:
1714:
1712:
1709:
1707:
1704:
1701:
1700:
1698:
1680:
1677:
1647:
1642:
1636:
1633:
1631:
1628:
1627:
1624:
1621:
1619:
1616:
1615:
1613:
1553:
1547:
1544:
1541:
1538:
1535:
1533:
1530:
1527:
1524:
1523:
1520:
1517:
1514:
1511:
1509:
1506:
1503:
1500:
1499:
1497:
1484:
1478:
1475:
1472:
1469:
1467:
1464:
1461:
1458:
1457:
1454:
1451:
1448:
1445:
1442:
1440:
1437:
1434:
1431:
1430:
1428:
1344:; then either
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1284:
1282:
1279:
1278:
1275:
1270:
1266:
1262:
1257:
1253:
1249:
1246:
1244:
1242:
1239:
1238:
1235:
1230:
1226:
1222:
1217:
1213:
1209:
1206:
1204:
1202:
1199:
1198:
1178:
1173:
1167:
1164:
1162:
1159:
1158:
1155:
1152:
1150:
1147:
1146:
1144:
1119:
1116:
1114:
1111:
1110:
1109:
1095:
1089:
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1070:
1067:
1064:
1062:
1059:
1057:
1054:
1052:
1049:
1048:
1045:
1042:
1040:
1037:
1035:
1032:
1030:
1027:
1026:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1004:
1002:
990:
972:
966:
963:
960:
958:
955:
954:
951:
948:
946:
943:
942:
940:
928:
914:
908:
905:
902:
899:
897:
894:
891:
888:
887:
884:
881:
878:
875:
872:
870:
867:
864:
861:
860:
858:
846:
832:
826:
823:
821:
818:
817:
814:
811:
809:
806:
805:
803:
786:
783:
715:
710:
705:
699:
694:
687:
680:
674:
671:
666:
661:
658:
652:
647:
641:
636:
633:
630:
625:
617:
610:
544:
538:
533:
529:
525:
521:
515:
510:
506:
502:
497:
492:
487:
433:
430:
379:rotoreflection
258:
253:
250:
246:
242:
236:
231:
177:
174:
171:
165:
160:
156:
153:
150:
144:
139:
105:linear algebra
99:
98:
53:
51:
44:
36:unitary matrix
32:complex number
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7573:
7562:
7559:
7558:
7556:
7541:
7538:
7536:
7533:
7531:
7530:
7525:
7519:
7518:
7515:
7509:
7506:
7504:
7501:
7499:
7498:Pseudoinverse
7496:
7494:
7491:
7489:
7486:
7484:
7481:
7479:
7476:
7474:
7471:
7470:
7468:
7466:Related terms
7464:
7458:
7457:Z (chemistry)
7455:
7453:
7450:
7448:
7445:
7443:
7440:
7438:
7435:
7433:
7430:
7428:
7425:
7423:
7420:
7418:
7415:
7413:
7410:
7408:
7405:
7403:
7400:
7398:
7395:
7394:
7392:
7388:
7382:
7379:
7377:
7374:
7372:
7369:
7367:
7364:
7362:
7359:
7357:
7354:
7352:
7349:
7347:
7344:
7343:
7341:
7339:
7334:
7328:
7325:
7323:
7320:
7318:
7315:
7313:
7310:
7308:
7305:
7303:
7300:
7298:
7295:
7293:
7290:
7288:
7285:
7283:
7280:
7279:
7277:
7275:
7270:
7264:
7261:
7259:
7256:
7254:
7251:
7249:
7246:
7244:
7241:
7239:
7236:
7234:
7231:
7229:
7226:
7224:
7221:
7219:
7216:
7214:
7211:
7209:
7206:
7204:
7201:
7199:
7196:
7194:
7191:
7189:
7186:
7184:
7181:
7179:
7176:
7174:
7171:
7169:
7166:
7164:
7161:
7159:
7156:
7154:
7151:
7149:
7146:
7144:
7141:
7139:
7136:
7134:
7131:
7129:
7126:
7124:
7121:
7119:
7116:
7114:
7111:
7110:
7108:
7104:
7098:
7095:
7093:
7090:
7088:
7085:
7083:
7080:
7078:
7075:
7073:
7070:
7068:
7065:
7063:
7060:
7058:
7055:
7053:
7050:
7048:
7044:
7041:
7039:
7036:
7035:
7033:
7031:
7027:
7022:
7016:
7013:
7011:
7008:
7006:
7003:
7001:
6998:
6996:
6993:
6991:
6988:
6986:
6983:
6981:
6978:
6977:
6975:
6973:
6968:
6962:
6959:
6957:
6954:
6952:
6949:
6947:
6944:
6942:
6939:
6937:
6934:
6932:
6929:
6927:
6924:
6922:
6919:
6917:
6914:
6913:
6911:
6907:
6901:
6898:
6896:
6893:
6891:
6888:
6886:
6883:
6881:
6878:
6876:
6873:
6871:
6868:
6866:
6863:
6861:
6858:
6856:
6853:
6851:
6848:
6846:
6843:
6841:
6838:
6836:
6833:
6831:
6828:
6826:
6823:
6821:
6818:
6816:
6815:Pentadiagonal
6813:
6811:
6808:
6806:
6803:
6801:
6798:
6796:
6793:
6791:
6788:
6786:
6783:
6781:
6778:
6776:
6773:
6771:
6768:
6766:
6763:
6761:
6758:
6756:
6753:
6751:
6748:
6746:
6743:
6741:
6738:
6736:
6733:
6731:
6728:
6726:
6723:
6721:
6718:
6716:
6713:
6711:
6708:
6706:
6703:
6701:
6698:
6696:
6693:
6691:
6688:
6686:
6683:
6681:
6678:
6676:
6673:
6671:
6668:
6666:
6663:
6661:
6658:
6656:
6653:
6651:
6648:
6646:
6645:Anti-diagonal
6643:
6641:
6638:
6637:
6635:
6631:
6626:
6619:
6614:
6612:
6607:
6605:
6600:
6599:
6596:
6590:
6587:
6583:
6579:
6578:
6573:
6569:
6568:
6564:
6559:
6555:
6550:
6545:
6541:
6537:
6532:
6529:
6525:
6521:
6517:
6513:
6509:
6505:
6501:
6497:
6493:
6490:
6486:
6482:
6478:
6474:
6470:
6466:
6462:
6458:
6454:
6452:
6449:
6445:
6441:
6437:
6433:
6429:
6424:
6419:
6415:
6411:
6407:
6403:
6400:
6396:
6392:
6388:
6384:
6380:
6373:
6369:
6365:
6362:
6356:
6352:
6348:
6344:
6340:
6336:
6332:
6328:
6323:
6320:
6316:
6312:
6308:
6304:
6300:
6296:
6292:
6288:
6284:
6283:
6279:
6271:
6267:
6264:
6259:
6256:
6252:
6248:
6244:
6239:
6236:
6232:
6228:
6223:
6220:
6213:
6209:
6206:
6205:
6201:
6199:
6197:
6193:
6188:
6184:
6180:For the case
6178:
6174:
6171:
6167:
6157:
6153:
6147:
6143:
6137:
6133:
6124:
6119:
6115:
6112:
6106:
6102:
6096:
6092:
6089:
6078:
6070:
6068:
6066:
6061:
6059:
6045:
6037:
6029:
6024:
6020:
6014:
6006:
6000:
5996:
5992:
5988:
5982:
5976:
5968:
5966:
5951:
5947:
5943:
5940:
5935:
5931:
5925:
5921:
5917:
5912:
5908:
5904:
5901:
5896:
5893:
5890:
5886:
5864:
5860:
5854:
5850:
5844:
5841:
5836:
5831:
5827:
5805:
5801:
5788:
5784:
5780:
5775:
5771:
5762:
5759:
5753:
5748:
5745:
5738:
5719:
5716:
5711:
5705:
5701:
5689:
5685:
5682:
5677:
5674:
5669:
5665:
5660:
5655:
5652:
5649:
5644:
5641:
5638:
5634:
5624:
5607:
5604:
5599:
5594:
5590:
5579:
5574:
5569:
5566:
5563:
5555:
5543:
5539:
5523:
5521:
5517:
5511:
5507:
5500:
5496:
5490:
5486:
5482:
5481:Mezzadri 2006
5474:
5470:
5469:decomposition
5463:
5459:
5455:
5451:
5444:Randomization
5443:
5441:
5426:
5420:
5415:
5408:
5405:
5400:
5394:
5384:
5378:
5373:
5366:
5361:
5355:
5345:
5342:
5327:
5321:
5316:
5309:
5306:
5301:
5295:
5285:
5279:
5274:
5267:
5264:
5259:
5253:
5243:
5237:
5232:
5225:
5220:
5214:
5204:
5184:
5178:
5173:
5166:
5163:
5158:
5152:
5144:
5136:
5130:
5125:
5118:
5113:
5107:
5097:
5091:
5086:
5079:
5074:
5068:
5057:
5055:
5051:
5047:
5046:Higham (1986)
5043:
5039:
5035:
5031:
5027:
5026:orthogonalize
5023:
5006:
5003:
5000:
4996:
4990:
4980:
4976:
4971:
4970:pseudoinverse
4966:
4962:
4953:
4948:
4945:
4942:
4936:
4932:
4929:
4925:
4922:
4917:
4913:
4904:
4901:
4895:
4891:
4885:
4882:
4876:
4873:
4869:
4866:
4863:
4845:
4837:
4833:
4830:
4822:
4817:
4812:
4799:
4794:
4788:
4783:
4778:
4771:
4766:
4761:
4754:
4749:
4744:
4737:
4732:
4727:
4720:
4715:
4710:
4704:
4699:
4696:
4689:has the form
4663:
4659:
4653:
4649:
4639:
4635:
4632:
4627:
4619:
4605:
4601:
4597:
4595:
4592:
4576:
4572:
4568:
4564:
4561:
4557:
4554:
4538:
4534:
4530:
4526:
4524:
4521:
4509:
4505:
4501:
4499:
4498:decomposition
4493:
4492:
4491:
4489:
4485:
4477:
4475:
4473:
4469:
4460:
4455:
4449:
4443:
4439:
4434:
4432:
4428:
4424:
4419:
4412:
4407:
4405:
4392:
4387:
4379:
4375:
4369:
4365:
4361:
4358:
4353:
4349:
4345:
4342:
4339:
4334:
4331:
4328:
4325:
4322:
4317:
4313:
4309:
4306:
4303:
4298:
4295:
4292:
4289:
4286:
4281:
4277:
4273:
4270:
4267:
4260:
4257:
4254:
4251:
4248:
4243:
4239:
4235:
4232:
4229:
4222:
4218:
4212:
4208:
4204:
4201:
4196:
4192:
4188:
4185:
4182:
4177:
4174:
4171:
4168:
4165:
4160:
4156:
4152:
4149:
4146:
4139:
4136:
4133:
4130:
4127:
4122:
4118:
4114:
4111:
4108:
4103:
4100:
4097:
4094:
4091:
4086:
4082:
4078:
4075:
4072:
4065:
4061:
4055:
4051:
4047:
4044:
4039:
4035:
4031:
4028:
4025:
4019:
4014:
4002:
3999:
3980:
3972:
3956:
3948:
3938:
3932:
3919:
3914:
3908:
3903:
3900:
3895:
3892:
3889:
3882:
3879:
3876:
3871:
3866:
3863:
3856:
3853:
3848:
3845:
3842:
3837:
3831:
3826:
3814:
3807:
3803:
3799:
3795:
3788:
3784:
3780:
3776:
3752:
3731:
3726:
3724:
3720:
3716:
3711:
3698:
3692:
3689:
3683:
3680:
3666:
3663:
3650:
3646:
3639:
3633:
3620:
3617:
3611:
3608:
3594:
3590:
3587:
3573:
3570:
3546:
3543:
3540:
3529:
3519:
3515:
3508:
3491:
3489:
3479:
3475:
3471:
3470:complex plane
3465:
3461:
3454:
3450:The matrices
3448:
3435:
3430:
3420:
3417:
3407:
3397:
3394:
3385:
3378:
3367:
3363:
3352:
3340:
3336:
3325:
3320:
3317:
3314:
3303:
3289:
3277:
3273:
3266:
3260:
3247:
3236:
3225:
3219:
3206:
3202:
3190:
3177:
3173:
3166:
3161:
3158:
3155:
3144:
3137:
3126:
3115:
3107:
3103:
3092:
3080:
3076:
3069:
3064:
3061:
3058:
3047:
3038:
3032:
3016:
3014:
3012:
2997:
2985:
2976:
2967:
2961:
2958:
2955:
2948:
2944:
2924:
2917:
2914:
2911:
2905:
2899:
2896:
2893:
2890:
2887:
2881:
2878:
2875:
2869:
2863:
2860:
2857:
2844:
2837:
2833:
2826:
2820:
2816:
2812:
2806:
2800:
2792:
2785:
2781:
2774:
2769:
2763:
2755:
2747:
2741:
2737:
2732:
2717:
2711:
2706:
2701:
2696:
2689:
2679:
2670:
2654:
2645:
2635:
2631:
2623:
2616:
2612:
2604:
2600:
2595:Now consider
2593:
2591:
2584:matrices. If
2573:
2567:
2561:
2555:
2551:
2541:
2537:
2532:
2526:
2521:
2517:
2511:
2505:
2502:
2497:
2493:
2487:
2484:, called the
2471:
2467:
2459:of dimension
2458:
2455:
2451:
2446:
2442:
2433:
2431:
2429:
2425:
2421:
2417:
2412:
2410:
2405:
2390:
2382:
2379:
2372:
2365:
2360:
2354:
2343:
2330:
2325:
2310:
2294:
2288:
2278:
2267:
2263:
2256:
2252:
2248:
2237:
2232:
2225:
2219:
2210:
2207:
2199:
2194:
2192:
2183:
2179:
2176:
2170:
2165:
2160:
2156:
2152:
2148:
2140:
2135:
2133:
2127:
2122:
2120:
2096:
2092:
2082:
2078:
2073:
2068:
2061:
2057:
2051:
2047:
2043:
2037:
2031:
2025:
2019:
2013:
2000:
1943:
1940:
1937:
1934:
1931:
1922:
1917:
1912:
1908:
1902:
1898:
1889:
1887:
1884:
1882:
1878:
1869:
1863:
1862:
1861:rotoinversion
1857:
1856:
1852:represent an
1850:
1835:
1829:
1826:
1821:
1816:
1809:
1804:
1799:
1792:
1787:
1784:
1779:
1773:
1761:
1755:
1752:
1747:
1742:
1735:
1730:
1727:
1722:
1715:
1710:
1705:
1702:
1696:
1678:
1676:
1674:
1670:
1666:
1661:
1658:
1645:
1640:
1634:
1629:
1622:
1617:
1611:
1601:
1588:
1584:
1577:
1571:
1551:
1545:
1542:
1539:
1536:
1531:
1528:
1525:
1518:
1515:
1512:
1507:
1504:
1501:
1495:
1482:
1476:
1473:
1470:
1465:
1462:
1459:
1452:
1449:
1446:
1443:
1438:
1435:
1432:
1426:
1416:
1403:
1392:
1382:
1378:
1372:
1368:
1362:
1358:
1352:
1348:
1342:
1338:
1332:
1328:
1322:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1285:
1280:
1273:
1268:
1264:
1260:
1255:
1251:
1247:
1245:
1240:
1233:
1228:
1224:
1220:
1215:
1211:
1207:
1205:
1200:
1176:
1171:
1165:
1160:
1153:
1148:
1142:
1127:
1117:
1112:
1093:
1087:
1082:
1077:
1072:
1065:
1060:
1055:
1050:
1043:
1038:
1033:
1028:
1021:
1016:
1011:
1006:
1000:
991:
988:
970:
964:
961:
956:
949:
944:
938:
929:
912:
906:
903:
900:
895:
892:
889:
882:
879:
876:
873:
868:
865:
862:
856:
847:
830:
824:
819:
812:
807:
801:
792:
791:
790:
784:
782:
780:
776:
772:
771:decomposition
765:
759:
753:
749:
744:
740:
734:
731:
726:
713:
703:
692:
672:
659:
634:
628:
596:
593:
587:
584:
578:
573:
564:
542:
531:
527:
523:
519:
508:
504:
500:
490:
476:
467:
461:
456:
452:
448:
447:normal matrix
438:
431:
429:
427:
421:
416:
412:
406:
401:
396:
392:
386:
384:
380:
376:
372:
368:
364:
360:
359:inner product
356:
352:
348:
343:
339:
336:
331:
323:
319:
314:
308:
304:
299:
294:
290:
280:
273:
256:
251:
248:
244:
240:
229:
220:
211:
209:
197:
192:
175:
172:
169:
158:
154:
151:
148:
137:
127:
125:
122:
118:
117:square matrix
114:
110:
106:
95:
92:
84:
74:
70:
64:
63:
57:
52:
43:
42:
37:
33:
19:
7520:
7452:Substitution
7338:graph theory
7071:
6835:Quaternionic
6825:Persymmetric
6575:
6539:
6535:
6503:
6499:
6464:
6460:
6413:
6409:
6382:
6378:
6350:
6334:
6330:
6294:
6290:
6258:
6238:
6222:
6186:
6182:
6179:
6175:
6169:
6165:
6155:
6151:
6145:
6141:
6139:matrix with
6135:
6131:
6122:
6117:
6113:
6110:
6104:
6100:
6094:
6090:
6087:
6080:
6062:
6043:
6035:
6027:
6022:
6012:
6009:. Likewise,
6004:
5994:
5980:
5972:
5969:Spin and pin
5763:
5760:
5749:
5743:
5736:
5625:
5527:
5515:
5509:
5505:
5498:
5494:
5458:Haar measure
5447:
5346:
5343:
5205:
5058:
5033:
5007:
5001:
4998:
4994:
4988:
4978:
4974:
4964:
4960:
4958:factored as
4949:
4943:
4940:
4934:
4930:
4927:
4923:
4920:
4902:
4899:
4893:
4889:
4883:
4880:
4874:
4871:
4867:
4864:
4861:
4843:
4835:
4831:
4828:
4820:
4813:
4661:
4657:
4651:
4647:
4637:
4633:
4630:
4624:Consider an
4623:
4611:orthogonal,
4603:
4599:
4586:orthogonal,
4574:
4570:
4566:
4548:orthogonal,
4536:
4532:
4528:
4515:orthogonal,
4507:
4503:
4481:
4471:
4458:
4450:
4435:
4416:
3978:
3970:
3954:
3946:
3936:
3933:
3812:
3805:
3801:
3797:
3793:
3786:
3782:
3778:
3774:
3727:
3712:
3648:
3644:
3637:
3634:
3517:
3513:
3506:
3495:
3463:
3459:
3452:
3449:
3287:
3275:
3271:
3264:
3261:
3039:
3020:
2995:
2983:
2974:
2965:
2953:
2949:
2942:
2842:
2835:
2831:
2824:
2818:
2814:
2810:
2798:
2790:
2786:
2779:
2772:
2761:
2753:
2745:
2733:
2636:
2629:
2621:
2614:
2610:
2602:
2598:
2594:
2571:
2559:
2539:
2535:
2524:
2509:
2498:
2491:
2469:
2465:
2444:
2440:
2437:
2416:diagonalized
2413:
2406:
2344:
2195:
2181:
2177:
2174:
2168:
2158:
2144:
2123:
2119:Euler angles
2094:
2090:
2080:
2076:
2069:
2059:
2055:
2049:
2045:
2041:
2035:
2029:
2023:
2017:
2014:
1920:
1913:
1906:
1900:
1896:
1893:
1885:
1870:
1859:
1853:
1851:
1682:
1662:
1659:
1586:
1582:
1575:
1572:
1417:
1401:
1390:
1380:
1376:
1370:
1366:
1360:
1356:
1350:
1346:
1340:
1336:
1330:
1326:
1323:
1128:
1121:
986:
788:
757:
742:
738:
735:
727:
594:
591:
585:
582:
576:
562:
465:
459:
455:dot products
444:
419:
404:
394:
390:
387:
369:, such as a
347:real numbers
341:
337:
334:
312:
306:
302:
292:
288:
281:
271:
212:
190:
128:
115:, is a real
112:
108:
102:
87:
78:
59:
7427:Hamiltonian
7351:Biadjacency
7287:Correlation
7203:Householder
7153:Commutation
6890:Vandermonde
6885:Tridiagonal
6820:Permutation
6810:Nonnegative
6795:Matrix unit
6675:Bisymmetric
6058:quaternions
5462:independent
5038:matrix norm
4887:is square (
4855:(and hence
4582:symmetric,
3768:tangent to
3715:Lie algebra
3504:, and that
3492:Lie algebra
3474:eigenvalues
2787:Similarly,
2776:with fiber
2424:eigenvalues
2198:determinant
2164:dot product
764:point group
754:denoted by
388:The set of
351:determinant
345:) over the
121:orthonormal
73:introducing
34:field, see
7327:Transition
7322:Stochastic
7292:Covariance
7274:statistics
7253:Symplectic
7248:Similarity
7077:Unimodular
7072:Orthogonal
7057:Involutory
7052:Invertible
7047:Projection
7043:Idempotent
6985:Convergent
6880:Triangular
6830:Polynomial
6775:Hessenberg
6745:Equivalent
6740:Elementary
6720:Copositive
6710:Conference
6670:Bidiagonal
6280:References
6019:pin groups
5999:spin group
3944:; setting
3131: even
2452:. It is a
2136:Properties
1890:Primitives
1598:; it is a
375:reflection
56:references
7508:Wronskian
7432:Irregular
7422:Gell-Mann
7371:Laplacian
7366:Incidence
7346:Adjacency
7317:Precision
7282:Centering
7188:Generator
7158:Confusion
7143:Circulant
7123:Augmented
7082:Unipotent
7062:Nilpotent
7038:Congruent
7015:Stieltjes
6990:Defective
6980:Companion
6951:Redheffer
6870:Symmetric
6865:Sylvester
6840:Signature
6770:Hermitian
6750:Frobenius
6660:Arrowhead
6640:Alternant
6582:EMS Press
6528:0036-1429
6489:120372682
6481:0029-599X
6440:0196-5204
6418:CiteSeerX
6399:0196-5204
6319:122752374
6311:0269-9648
6297:: 15–32,
5985:, is not
5975:connected
5941:−
5717:−
5675:−
5600:−
5406:−
5390:→
5307:−
5291:→
5265:−
5249:→
5164:−
5148:→
5145:⋯
5142:→
5103:→
4755:⋅
4738:⋅
4733:⋅
4721:⋅
4716:⋅
4711:⋅
4456:of order
4448:indices.
4343:−
4287:−
4249:−
4186:−
4092:−
4029:−
4009:Ω
4003:
3940:by angle
3904:θ
3896:θ
3890:−
3883:θ
3877:−
3867:θ
3857:θ
3849:θ
3843:−
3824:Ω
3693:˙
3684:−
3667:˙
3612:˙
3574:˙
3418:±
3408:⋱
3395:±
3353:⋱
3241: odd
3191:⋱
3093:⋱
2915:−
2891:⋯
2879:−
2861:−
2702:⋯
2680:⋮
2457:Lie group
2430: 1.
2418:over the
2409:signature
1941:−
1855:inversion
1827:−
1785:−
1753:−
1728:−
1703:−
1669:symmetric
1546:θ
1543:
1537:−
1532:θ
1529:
1519:θ
1516:
1508:θ
1505:
1477:θ
1474:
1466:θ
1463:
1453:θ
1450:
1444:−
1439:θ
1436:
962:−
907:θ
904:
896:θ
893:
883:θ
880:
874:−
869:θ
866:
777:(used in
524:⋅
491:⋅
310:), where
249:−
196:transpose
7561:Matrices
7555:Category
7336:Used in
7272:Used in
7233:Rotation
7208:Jacobian
7168:Distance
7148:Cofactor
7133:Carleman
7113:Adjugate
7097:Weighing
7030:inverses
7026:products
6995:Definite
6926:Identity
6916:Exchange
6909:Constant
6875:Toeplitz
6760:Hadamard
6730:Diagonal
6448:14268409
6370:(1986),
6349:(1996),
6266:Archived
6202:See also
5497:+ 1) × (
5421:0.393919
5416:0.919145
5409:0.919145
5401:0.393919
4982:, where
4839:‖
4826:‖
4641:, where
4620:Examples
4590:diagonal
4470:, we do
4413:Benefits
3772:. Given
2601:+ 1) × (
785:Examples
432:Overview
415:subgroup
371:rotation
363:isometry
81:May 2023
7437:Overlap
7402:Density
7361:Edmonds
7238:Seifert
7198:Hessian
7163:Coxeter
7087:Unitary
7005:Hurwitz
6936:Of ones
6921:Hilbert
6855:Skyline
6800:Metzler
6790:Logical
6785:Integer
6695:Boolean
6627:classes
6584:, 2001
6554:Bibcode
6508:Bibcode
6337:: 21–25
6050:Spin(3)
6025:). For
5997:), the
5280:1.41421
5275:1.06066
5268:1.06066
5260:1.41421
5034:closest
3989:
3975:
3965:
3951:
3791:, with
3559:yields
3458:, ...,
3270:, ...,
3007:
2992:
2817:+ 1) →
2813:) ↪ SO(
2518:2, the
2481:
2462:
2454:compact
2428:modulus
2185:, with
2153:of the
2106:
2087:
1868:-axis.
1412:
1398:
1388:(where
316:is the
298:unitary
219:inverse
206:is the
194:is the
124:vectors
69:improve
7356:Degree
7297:Design
7228:Random
7218:Payoff
7213:Moment
7138:Cartan
7128:Bézout
7067:Normal
6941:Pascal
6931:Lehmer
6860:Sparse
6780:Hollow
6765:Hankel
6700:Cauchy
6625:Matrix
6526:
6487:
6479:
6446:
6438:
6420:
6397:
6357:
6317:
6309:
6129:is an
6030:> 2
6021:, Pin(
5993:of SO(
5734:where
5131:2.6875
5126:3.4375
5119:0.0625
5114:1.8125
5024:could
4878:. Now
3973:= sin
3949:= cos
3511:gives
3480:1. If
3231:
3141:
3121:
2768:bundle
2554:splits
2033:). If
1339:= sin
1329:= cos
989:-axis)
566:in an
556:where
469:in an
413:. The
349:. The
330:normal
269:where
188:where
58:, but
7417:Gamma
7381:Tutte
7243:Shear
6956:Shift
6946:Pauli
6895:Walsh
6805:Moore
6685:Block
6544:arXiv
6485:S2CID
6444:S2CID
6375:(PDF)
6315:S2CID
6214:Notes
6054:SU(2)
6034:Spin(
6003:Spin(
5014:3 × 3
5012:is a
4685:then
4683:5 × 3
4660:>
4440:with
3770:SO(3)
3486:3 × 3
3476:have
3293:2 × 2
3282:2 × 2
2766:is a
2582:2 × 2
2564:is a
2550:coset
2538:)/SO(
2516:index
2450:group
2130:2 × 2
2115:3 × 3
2111:3 × 3
1873:3 × 3
1685:3 × 3
1578:= 90°
1131:2 × 2
1124:1 × 1
748:group
728:Thus
451:field
400:group
324:) of
111:, or
107:, an
7223:Pick
7193:Gram
6961:Zero
6665:Band
6524:ISSN
6477:ISSN
6436:ISSN
6395:ISSN
6355:ISBN
6307:ISSN
6098:and
5501:+ 1)
5050:1990
4814:The
4665:. A
4544:and
3280:are
2801:+ 1)
2764:+ 1)
2756:+ 1)
2632:+ 1)
2605:+ 1)
2578:O(1)
2546:O(1)
2472:− 1)
2196:The
2097:− 1)
1594:and
1129:The
463:and
202:and
7312:Hat
7045:or
7028:or
6516:doi
6469:doi
6428:doi
6387:doi
6299:doi
6251:MIT
6081:If
6042:SO(
5979:SO(
5754:of
5544:of
5518:+ 1
5483:).
5322:0.8
5317:0.6
5310:0.6
5302:0.8
5179:0.8
5174:0.6
5167:0.6
5159:0.8
4992:to
4938:to
4926:= (
4681:is
4645:is
4472:not
4000:exp
3816:is
3796:= (
3777:= (
3640:= 0
3509:= 0
2986:+ 1
2841:SO(
2827:− 1
2809:SO(
2797:SO(
2789:SO(
2750:in
2576:by
2570:SO(
2568:of
2523:SO(
2514:of
2506:of
2305:det
2283:det
2260:det
2229:det
2214:det
2048:− 2
1924:as
1909:− 1
1540:cos
1526:sin
1513:sin
1502:cos
1471:cos
1460:sin
1447:sin
1433:cos
1393:= 0
1379:= −
1364:or
1349:= −
901:cos
890:sin
877:sin
863:cos
779:MP3
580:is
418:SO(
377:or
365:of
296:),
198:of
103:In
7557::
6580:,
6574:,
6552:,
6542:,
6540:54
6538:,
6522:,
6514:,
6504:17
6502:,
6483:,
6475:,
6465:25
6463:,
6442:,
6434:,
6426:,
6414:11
6412:,
6393:,
6381:,
6377:,
6345:;
6333:,
6329:,
6313:,
6305:,
6293:,
6249:,
6245:,
6198:.
6185:≤
6173:.
6168:≥
6154:=
6152:QQ
6144:≤
6134:×
6116:=
6103:=
6101:QQ
6093:=
6060:.
6032:,
6011:O(
6001:,
5747:.
5742:=
5556::
5508:×
5467:QR
5005:.
4972:,
4935:QR
4892:×
4870:=
4834:−
4667:QR
4655:,
4650:×
4636:=
4607:,
4604:QS
4602:=
4578:,
4569:=
4540:,
4531:=
4511:,
4508:QR
4506:=
4496:QR
3992:,
3968:,
3804:,
3800:,
3787:zθ
3785:,
3783:yθ
3781:,
3779:xθ
3647:=
3516:=
3013:.
2947:.
2941:O(
2834:×
2784:.
2778:O(
2760:O(
2752:O(
2744:O(
2628:O(
2620:O(
2613:×
2558:O(
2556:,
2534:O(
2508:O(
2496:.
2490:O(
2443:×
2193:.
2189:a
2180:=
2124:A
2121:.
2079:×
2070:A
2058:×
2050:vv
2044:=
1914:A
1899:×
1883:.
1585:=
1415:.
1374:,
1369:=
1359:=
1354:,
1334:,
769:QR
756:O(
741:×
441:A.
403:O(
393:×
385:.
373:,
342:QQ
340:=
305:=
291:=
279:.
221::
210:.
126:.
7442:S
6900:Z
6617:e
6610:t
6603:v
6556::
6546::
6518::
6510::
6471::
6430::
6389::
6383:7
6335:8
6301::
6295:1
6253:.
6187:m
6183:n
6170:m
6166:n
6161:Q
6156:I
6146:m
6142:n
6136:n
6132:m
6127:Q
6123:Q
6118:I
6114:Q
6111:Q
6105:I
6095:I
6091:Q
6088:Q
6083:Q
6046:)
6044:n
6038:)
6036:n
6028:n
6023:n
6015:)
6013:n
6007:)
6005:n
5995:n
5983:)
5981:n
5952:n
5948:P
5944:3
5936:n
5932:N
5926:n
5922:P
5918:+
5913:n
5909:Q
5905:2
5902:=
5897:1
5894:+
5891:n
5887:Q
5865:n
5861:N
5855:n
5851:Q
5845:2
5842:1
5837:=
5832:n
5828:P
5806:n
5802:Q
5795:T
5789:n
5785:Q
5781:=
5776:n
5772:N
5756:M
5744:M
5740:0
5737:Q
5720:1
5712:)
5706:n
5702:Q
5695:T
5690:M
5686:+
5683:M
5678:1
5670:n
5666:Q
5661:(
5656:M
5653:2
5650:=
5645:1
5642:+
5639:n
5635:Q
5608:2
5605:1
5595:)
5591:M
5585:T
5580:M
5575:(
5570:M
5567:=
5564:Q
5550:R
5546:M
5534:M
5530:Q
5516:n
5510:n
5506:n
5499:n
5495:n
5493:(
5477:R
5427:]
5395:[
5385:]
5379:5
5374:7
5367:1
5362:3
5356:[
5328:]
5296:[
5286:]
5254:[
5244:]
5238:5
5233:7
5226:1
5221:3
5215:[
5201:γ
5185:]
5153:[
5137:]
5108:[
5098:]
5092:5
5087:7
5080:1
5075:3
5069:[
5048:(
5018:A
5010:A
5002:b
4999:U
4997:Σ
4995:V
4989:x
4984:Σ
4979:U
4977:Σ
4975:V
4965:V
4963:Σ
4961:U
4956:A
4944:R
4941:R
4933:)
4931:Q
4928:R
4924:A
4921:A
4914:(
4908:R
4903:R
4900:R
4894:n
4890:n
4884:A
4881:A
4875:b
4872:A
4868:x
4865:A
4862:A
4857:R
4853:A
4849:A
4844:b
4836:b
4832:x
4829:A
4821:x
4800:.
4795:]
4789:0
4784:0
4779:0
4772:0
4767:0
4762:0
4750:0
4745:0
4728:0
4705:[
4700:=
4697:R
4687:R
4679:A
4675:R
4671:A
4662:n
4658:m
4652:n
4648:m
4643:A
4638:b
4634:x
4631:A
4613:S
4609:Q
4600:M
4588:Λ
4584:Q
4580:S
4575:Q
4573:Λ
4571:Q
4567:S
4562:)
4550:Σ
4546:V
4542:U
4537:V
4535:Σ
4533:U
4529:M
4517:R
4513:Q
4504:M
4486:(
4464:n
4459:n
4446:n
4393:.
4388:]
4380:2
4376:s
4370:2
4366:z
4362:2
4359:+
4354:2
4350:s
4346:2
4340:1
4335:c
4332:s
4329:x
4326:2
4323:+
4318:2
4314:s
4310:z
4307:y
4304:2
4299:c
4296:s
4293:y
4290:2
4282:2
4278:s
4274:z
4271:x
4268:2
4261:c
4258:s
4255:x
4252:2
4244:2
4240:s
4236:z
4233:y
4230:2
4223:2
4219:s
4213:2
4209:y
4205:2
4202:+
4197:2
4193:s
4189:2
4183:1
4178:c
4175:s
4172:z
4169:2
4166:+
4161:2
4157:s
4153:y
4150:x
4147:2
4140:c
4137:s
4134:y
4131:2
4128:+
4123:2
4119:s
4115:z
4112:x
4109:2
4104:c
4101:s
4098:z
4095:2
4087:2
4083:s
4079:y
4076:x
4073:2
4066:2
4062:s
4056:2
4052:x
4048:2
4045:+
4040:2
4036:s
4032:2
4026:1
4020:[
4015:=
4012:)
4006:(
3986:2
3983:/
3979:θ
3971:s
3962:2
3959:/
3955:θ
3947:c
3942:θ
3937:v
3920:.
3915:]
3909:0
3901:x
3893:y
3880:x
3872:0
3864:z
3854:y
3846:z
3838:0
3832:[
3827:=
3813:ω
3808:)
3806:z
3802:y
3798:x
3794:v
3789:)
3775:ω
3756:)
3753:3
3750:(
3745:o
3742:s
3699:.
3690:Q
3681:=
3675:T
3664:Q
3649:I
3645:Q
3642:(
3638:t
3621:0
3618:=
3609:Q
3600:T
3595:Q
3591:+
3588:Q
3582:T
3571:Q
3547:I
3544:=
3541:Q
3535:T
3530:Q
3518:I
3514:Q
3507:t
3502:t
3498:Q
3482:n
3464:k
3460:R
3456:1
3453:R
3436:,
3431:]
3421:1
3398:1
3386:0
3379:0
3368:k
3364:R
3341:1
3337:R
3326:[
3321:=
3318:P
3315:Q
3309:T
3304:P
3288:I
3286:±
3276:k
3272:R
3268:1
3265:R
3248:.
3245:)
3237:n
3234:(
3226:]
3220:1
3207:k
3203:R
3178:1
3174:R
3167:[
3162:=
3159:P
3156:Q
3150:T
3145:P
3138:,
3135:)
3127:n
3124:(
3116:]
3108:k
3104:R
3081:1
3077:R
3070:[
3065:=
3062:P
3059:Q
3053:T
3048:P
3035:Q
3027:P
3023:Q
3004:2
3001:/
2998:!
2996:n
2984:n
2981:S
2975:n
2972:S
2966:n
2963:S
2956:!
2954:n
2945:)
2943:n
2925:2
2921:)
2918:1
2912:n
2909:(
2906:n
2900:=
2897:1
2894:+
2888:+
2885:)
2882:2
2876:n
2873:(
2870:+
2867:)
2864:1
2858:n
2855:(
2845:)
2843:n
2836:n
2832:n
2825:n
2819:S
2815:n
2811:n
2799:n
2793:)
2791:n
2782:)
2780:n
2773:S
2762:n
2754:n
2748:)
2746:n
2718:]
2712:1
2707:0
2697:0
2690:0
2674:)
2671:n
2668:(
2664:O
2655:0
2646:[
2630:n
2624:)
2622:n
2615:n
2611:n
2603:n
2599:n
2597:(
2586:n
2574:)
2572:n
2562:)
2560:n
2542:)
2540:n
2536:n
2527:)
2525:n
2512:)
2510:n
2494:)
2492:n
2478:2
2475:/
2470:n
2468:(
2466:n
2445:n
2441:n
2391:]
2383:2
2380:1
2373:0
2366:0
2361:2
2355:[
2331:.
2326:2
2320:)
2314:)
2311:Q
2308:(
2300:(
2295:=
2292:)
2289:Q
2286:(
2279:)
2273:T
2268:Q
2264:(
2257:=
2253:)
2249:Q
2243:T
2238:Q
2233:(
2226:=
2223:)
2220:I
2217:(
2211:=
2208:1
2187:D
2182:D
2178:M
2175:M
2169:R
2159:R
2103:2
2100:/
2095:n
2093:(
2091:n
2081:n
2077:n
2065:n
2060:n
2056:n
2046:I
2042:Q
2036:v
2030:v
2024:v
2018:v
2001:.
1993:v
1985:T
1978:v
1967:T
1960:v
1952:v
1944:2
1938:I
1935:=
1932:Q
1921:v
1907:n
1901:n
1897:n
1866:z
1836:]
1830:1
1822:0
1817:0
1810:0
1805:0
1800:1
1793:0
1788:1
1780:0
1774:[
1762:]
1756:1
1748:0
1743:0
1736:0
1731:1
1723:0
1716:0
1711:0
1706:1
1697:[
1646:.
1641:]
1635:0
1630:1
1623:1
1618:0
1612:[
1596:y
1592:x
1587:x
1583:y
1576:θ
1552:]
1496:[
1483:]
1427:[
1409:2
1406:/
1402:θ
1391:θ
1386:θ
1381:p
1377:u
1371:q
1367:t
1361:p
1357:u
1351:q
1347:t
1341:θ
1337:q
1331:θ
1327:p
1306:.
1303:u
1300:t
1297:+
1294:q
1291:p
1288:=
1281:0
1274:,
1269:2
1265:u
1261:+
1256:2
1252:q
1248:=
1241:1
1234:,
1229:2
1225:t
1221:+
1216:2
1212:p
1208:=
1201:1
1177:,
1172:]
1166:u
1161:q
1154:t
1149:p
1143:[
1094:]
1088:0
1083:0
1078:1
1073:0
1066:0
1061:0
1056:0
1051:1
1044:0
1039:1
1034:0
1029:0
1022:1
1017:0
1012:0
1007:0
1001:[
987:x
971:]
965:1
957:0
950:0
945:1
939:[
913:]
857:[
831:]
825:1
820:0
813:0
808:1
802:[
760:)
758:n
743:n
739:n
714:.
709:v
704:Q
698:T
693:Q
686:T
679:v
673:=
670:)
665:v
660:Q
657:(
651:T
646:)
640:v
635:Q
632:(
629:=
624:v
616:T
609:v
595:v
592:Q
586:v
583:v
577:v
568:n
563:v
558:Q
543:)
537:v
532:Q
528:(
520:)
514:u
509:Q
505:(
501:=
496:v
486:u
471:n
466:v
460:u
422:)
420:n
407:)
405:n
395:n
391:n
338:Q
335:Q
332:(
326:Q
320:(
313:Q
307:Q
303:Q
300:(
293:Q
289:Q
284:Q
277:Q
272:Q
257:,
252:1
245:Q
241:=
235:T
230:Q
215:Q
204:I
200:Q
191:Q
176:,
173:I
170:=
164:T
159:Q
155:Q
152:=
149:Q
143:T
138:Q
94:)
88:(
83:)
79:(
65:.
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.