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In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. [1] That is, the matrix A {\displaystyle A} is skew-Hermitian if it satisfies the relation
The difference of a square matrix and its conjugate transpose () is skew-Hermitian (also called antihermitian). This implies that the commutator of two Hermitian matrices is skew-Hermitian. An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B .
Every complex skew-Hermitian form can be written as the imaginary unit:= times a Hermitian form. The matrix representation of a complex skew-Hermitian form is a skew-Hermitian matrix. A complex skew-Hermitian form applied to a single vector | | = (,) is always a purely imaginary number.
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of + being , for real numbers and ).
Hankel matrix: A matrix with constant skew-diagonals; also an upside down Toeplitz matrix. A square Hankel matrix is symmetric. Hermitian matrix: A square matrix which is equal to its conjugate transpose, A = A *. Hessenberg matrix: An "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal.
The Lie algebra () of consists of n × n skew-Hermitian matrices with trace zero. [4] This (real) Lie algebra has dimension n 2 − 1. More information about the structure of this Lie algebra can be found below in § Lie algebra structure.
An matrix is said to be skew-symmetrizable if there exists an invertible diagonal matrix such that is skew-symmetric. For real n × n {\displaystyle n\times n} matrices, sometimes the condition for D {\displaystyle D} to have positive entries is added.
In particular all unitary, Hermitian, or skew-Hermitian (in the real-valued case, all orthogonal, symmetric, or skew-symmetric, respectively) matrices are normal and therefore possess this property. Comment: For any real symmetric matrix A , the eigendecomposition always exists and can be written as A = V D V T {\displaystyle A=VDV^{\mathsf {T ...