In mathematics, especially in linear algebra and matrix theory, a **centrosymmetric matrix** is a matrix which is symmetric about its center.

An *n* × *n* matrix *A* = [*A*_{i, j}] is centrosymmetric when its entries satisfy

Alternatively, if J denotes the *n* × *n* exchange matrix with 1 on the antidiagonal and 0 elsewhere:

then a matrix A is centrosymmetric if and only if

- If A and B are
*n*×*n*centrosymmetric matrices over a field F, then so are*A*+*B*and cA for any c in F. Moreover, the matrix product AB is centrosymmetric, since*JAB*=*AJB*=*ABJ*. Since the identity matrix is also centrosymmetric, it follows that the set of*n*×*n*centrosymmetric matrices over F forms a subalgebra of the associative algebra of all*n*×*n*matrices. - If A is a centrosymmetric matrix with an m-dimensional eigenbasis, then its m eigenvectors can each be chosen so that they satisfy either
**x**=*J***x**or**x**= −*J***x**where J is the exchange matrix. - If A is a centrosymmetric matrix with distinct eigenvalues, then the matrices that commute with A must be centrosymmetric.
^{[1]} - The maximum number of unique elements in an
*m*×*m*centrosymmetric matrix is

An *n* × *n* matrix A is said to be *skew-centrosymmetric* if its entries satisfy

Equivalently, A is skew-centrosymmetric if

The centrosymmetric relation *AJ* = *JA* lends itself to a natural generalization, where J is replaced with an involutory matrix K (i.e., *K*^{2} = *I* )^{[2]}^{[3]}^{[4]} or, more generally, a matrix K satisfying *K ^{m}* =

Symmetric centrosymmetric matrices are sometimes called bisymmetric matrices. When the ground field is the real numbers, it has been shown that bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix.^{[3]} A similar result holds for Hermitian centrosymmetric and skew-centrosymmetric matrices.^{[5]}