Centrosymmetric matrix

An n × n matrix A = [Ai, 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 AJ = JA.

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 )[2][3][4] or, more generally, a matrix K satisfying Km = I for an integer m > 1.

[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.

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.