mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That...

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...

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...

} 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...

, 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...

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...

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...

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...

}=\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...

theorem using skew-symmetric matrices and the tensor formulation, the following identities are useful. Let [R] be the skew symmetric matrix associated with...

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...

= 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...

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...

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...

antisymmetric (or skew-symmetric) is used for matrices, tensors, and other objects that change sign if an appropriate operation (e.g. matrix transposition)...

be equivalently written as matrix multiplication by combining the first operand and the operator into a skew-symmetric matrix, [ b ] {\displaystyle \left[\mathbf...

orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal...

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...

Infinite skew polyhedron Skew-symmetric graph Skew-symmetric matrix Skew tableau, a generalization of Young tableaux Skewness, a measure of the asymmetry...

described by Cayley (1846), the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography...

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...

_{\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...

respectively. Likewise, among real matrices, all orthogonal, symmetric, and skew-symmetric matrices are normal, with all eigenvalues being complex conjugate...

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...

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...

characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices". SIAM J. Matrix Anal. Appl. 23 (3): 885–895...

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...

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...

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...

Symmetric matrix Orthogonal matrix Skew-symmetric matrix Conjugate transpose Unitary matrix Hermitian matrix, Antihermitian matrix Positive-definite, positive-semidefinite...