mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That...
19 KB (3,564 words) - 08:34, 27 August 2024
a symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{ is symmetric}}\iff...
17 KB (2,812 words) - 05:04, 29 August 2024
the determinant of an m-by-m skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer...
22 KB (3,810 words) - 09:23, 17 August 2024
} is a fixed 2 n × 2 n {\displaystyle 2n\times 2n} nonsingular, skew-symmetric matrix. This definition can be extended to 2 n × 2 n {\displaystyle 2n\times...
15 KB (2,322 words) - 09:09, 14 August 2024
, is a symmetric matrix. If instead A T = − A {\displaystyle A^{\mathsf {T}}=-A} , then A {\displaystyle A} is called a skew-symmetric matrix. For a complex...
16 KB (1,831 words) - 06:07, 16 June 2024
complex conjugation. Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely...
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which maps any skew-symmetric matrix A to a rotation matrix. In fact, aside from the noted exceptions, we can produce any rotation matrix in this way. Although...
99 KB (15,019 words) - 17:05, 31 July 2024
special orthogonal group), the differential of a rotation is a skew-symmetric matrix A T = − A {\displaystyle A^{\mathsf {T}}=-A} in the tangent space...
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Transpose (redirect from Transpose of a matrix)
}=\mathbf {A} .} A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, A is skew-symmetric if A T = − A . {\displaystyle...
20 KB (2,525 words) - 12:30, 8 May 2024
theorem using skew-symmetric matrices and the tensor formulation, the following identities are useful. Let [R] be the skew symmetric matrix associated with...
12 KB (2,002 words) - 10:51, 30 April 2024
mathematics, a Hamiltonian matrix is a 2n-by-2n matrix A such that JA is symmetric, where J is the skew-symmetric matrix J = [ 0 n I n − I n 0 n ] {\displaystyle...
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= AT, is a symmetric matrix. If instead, A is equal to the negative of its transpose, that is, A = −AT, then A is a skew-symmetric matrix. In complex...
106 KB (13,107 words) - 02:49, 27 August 2024
Bilinear form (redirect from Skew-symmetric bilinear form)
any basis) is symmetric (respectively skew-symmetric). A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal...
22 KB (2,700 words) - 23:21, 30 May 2024
Angular velocity tensor (section Ω is skew-symmetric)
The angular velocity tensor is a skew-symmetric matrix defined by: Ω = ( 0 − ω z ω y ω z 0 − ω x − ω y ω x 0 ) {\displaystyle \Omega ={\begin{pmatrix}0&-\omega...
14 KB (2,523 words) - 15:45, 8 September 2023
Antisymmetric (redirect from Skew-symmetric)
antisymmetric (or skew-symmetric) is used for matrices, tensors, and other objects that change sign if an appropriate operation (e.g. matrix transposition)...
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Moment of inertia (redirect from Symmetric top)
be equivalently written as matrix multiplication by combining the first operand and the operator into a skew-symmetric matrix, [ b ] {\displaystyle \left[\mathbf...
91 KB (17,259 words) - 03:43, 6 August 2024
orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal...
36 KB (4,802 words) - 16:45, 22 May 2024
In linear algebra, a skew-Hamiltonian matrix is a specific type of matrix that corresponds to a skew-symmetric bilinear form on a symplectic vector space...
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Infinite skew polyhedron Skew-symmetric graph Skew-symmetric matrix Skew tableau, a generalization of Young tableaux Skewness, a measure of the asymmetry...
1 KB (176 words) - 20:59, 9 July 2024
Cayley transform (section Matrix map)
described by Cayley (1846), the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography...
14 KB (2,249 words) - 10:33, 28 June 2024
Cross product (redirect from Cross product matrix)
dimension bivectors can be identified with skew-symmetric matrices, so the product between a skew-symmetric matrix and vector is equivalent to the grade-1...
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_{\frac {n}{2}}} . Note that the (simple) graph of a real symmetric or skew-symmetric matrix A {\displaystyle A} of order n {\displaystyle n} has n {\displaystyle...
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respectively. Likewise, among real matrices, all orthogonal, symmetric, and skew-symmetric matrices are normal, with all eigenvalues being complex conjugate...
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be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A {\displaystyle A} is denoted by A H , {\displaystyle...
20 KB (3,013 words) - 18:34, 4 August 2024
as the matrix product of one rotation matrix and one skew-symmetric matrix, both 3 × 3 {\displaystyle 3\times 3} . The skew-symmetric matrix must have...
20 KB (3,215 words) - 14:23, 16 May 2024
characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices". SIAM J. Matrix Anal. Appl. 23 (3): 885–895...
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Sesquilinear form (redirect from Skew-Hermitian form)
2, skew-symmetric and symmetric bilinear forms coincide since then 1 = −1. In all cases, alternating bilinear forms are a subset of skew-symmetric bilinear...
23 KB (2,832 words) - 13:49, 2 February 2024
space. This is conventionally represented by a 3×3 skew-symmetric matrix A. It is not the matrix of an actual rotation in space; but for small real values...
4 KB (563 words) - 05:38, 17 May 2023
nonzero eigenvalues. Note that the (simple) graph of a real symmetric or skew-symmetric matrix A {\displaystyle A} of order n {\displaystyle n} has n {\displaystyle...
23 KB (2,938 words) - 07:47, 18 July 2024
Symmetric matrix Orthogonal matrix Skew-symmetric matrix Conjugate transpose Unitary matrix Hermitian matrix, Antihermitian matrix Positive-definite, positive-semidefinite...
5 KB (377 words) - 12:12, 30 October 2023