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

$${\displaystyle A_{i,\,j}=A_{n-i+1,\,n-j+1}\quad {\text{for all }}i,j\in \{1,\,\ldots ,\,n\}.}$$

Alternatively, if J denotes the n × n exchange matrix with 1 on the antidiagonal and 0 elsewhere: $${\displaystyle J_{i,\,j}={\begin{cases}1,&i+j=n+1\\0,&i+j\neq n+1\\\end{cases}}}$$ then a matrix A is centrosymmetric if and only if AJ = JA.

this is also the dimension of the vector space of all m × m centrosymmetric matrices

An n × n matrix A is said to be skew-centrosymmetric if its entries satisfy $${\displaystyle A_{i,\,j}=-A_{n-i+1,\,n-j+1}\quad {\text{for all }}i,j\in \{1,\,\ldots ,\,n\}.}$$ Equivalently, A is skew-centrosymmetric if AJ = −JA, where J is the exchange matrix defined previously.

The centrosymmetric relation AJ = JA lends itself to a natural generalization, where J is replaced with an involutory matrix K (i.e., K² = I ) or, more generally, a matrix K satisfying K^m = I for an integer m > 1. The inverse problem for the commutation relation AK = KA of identifying all involutory K that commute with a fixed matrix A has also been studied.

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. A similar result holds for Hermitian centrosymmetric and skew-centrosymmetric matrices.