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 = [Ai,j] is centrosymmetric when its entries satisfy Ai,j = An−i + 1,n−j + 1 for i, j ∊{1, ..., n}. If J denotes the n×n exchange matrix with 1 on the antidiagonal and 0 elsewhere (that is, Ji,n + 1 − i = 1; Ji,j = 0 if j ≠ n +1− i), then a matrix A is centrosymmetric if and only if AJ = JA. All 2×2 centrosymmetric matrices have the form All 3×3 centrosymmetric matrices have the form Symmetric Toeplitz matrices are centrosymmetric. If A and B are 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 is 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 = Jx or x = −Jx 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. The maximum number of unique elements in a m × m centrosymmetric matrix is . An n×n matrix A is said to be skew-centrosymmetric if its entries satisfy Ai,j = −An−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., K2 = I) or, more generally, a matrix K satisfying Km = 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.