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...

} 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...

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...

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...

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 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...

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...

= 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...

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

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

}=\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...

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...

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

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

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...

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

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

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...

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...

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...

gradient of a vector field is skew-symmetric at every point, then the gradient must be equal to a constant skew-symmetric matrix. Korn's theorem is a quantitative...

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...

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

the skew-symmetric matrix Hankel matrix, a square matrix in which each ascending skew-diagonal from left to right is constant Hasse–Witt matrix Hat matrix...

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...

_{\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...

Correlation matrix — a symmetric n×n matrix, formed by the pairwise correlation coefficients of several random variables. Covariance matrix — a symmetric n×n...

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