Centrosymmetric matrix

In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an "n" × "n" matrix "A" = ["A""i", "j"&hairsp;] is centrosymmetric when its entries satisfy
If "J" denotes the "n" × "n" matrix with 1 on the counterdiagonal and 0 elsewhere (that is, "J""i", "n"+1−"i" = 1; "J""i", "j" = 0 if "j" ≠ "n" + 1 − "i"), then a matrix "A" is centrosymmetric if and only if "AJ" = "JA". The matrix "J" is sometimes referred to as the exchange matrix.
Related structures.
An "n" × "n" matrix "A" is said to be "skew-centrosymmetric" if its entries satisfy "A""i", "j" = −"A""n"−"i"+1, "n"−"j"+1 for "i", "j" ∊ {1, …, "n"}. Equivalently, "A" is skew-centrosymmetric if "AJ" = −"JA", where "J" is the exchange matrix defined above.
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") or, more generally, a matrix "K" satisfying "Km = I" for an integer "m" > 1. The inverse problem for the commutation relation 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 field of 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.