The Inverse of a Centrosymmetric Matrix

A centrosymmetric matrix R = (R j,) (i, j = 1, 2, * * *, n) is one for which Rn+1-i,n+1-j = Ri,1 (i, i = 1, 2, * * *, n). An important special case of a centrosymmetric matrix is a Toeplitz matrix, which is a matrix for which all elements at each fixed perpendicular distance from the main diagonal are equal. It was shown by Ray (1969) that a finite Toeplitz matrix of order 2m or 2m + 1 can be inverted by inverting two m X m matrices, without essential matrix multiplications. (There are also some reversals of the rows or columns of matrices, but we do not regard these as "essential" matrix multiplications.) The main purpose of the present paper is to show that these results are valid also for centrosymmetric matrices, which include Toeplitz matrices as a special case. It is perhaps worth noting that both centrosymmetric matrices and Toeplitz matrices of a given order form Abelian (commutative) groups under addition; and that non-singular centrosymmetric matrices form a non-Abelian group under multiplication, whereas non-singular Toeplitz matrices do not form a group under multiplication since the product of two Toeplitz matrices is not usually a Toeplitz matrix. (For the definition of a group, see, for example, Ledermann, 1949.) Toeplitz matrices arise as discrete approximations to kernels k(x, t) of integral equations when these kernels are functions of Ix t\. (For integral equations see, for example, Smithies, 1958.) Similarly if a kernel is an even function of its vector argument (x, t), that is, if k(x, t) = k(-x, -t), then it can be discretely approximated by a centrosymmetric matrix. It would also be possible for x and t themselves to be vectors, and then the following theory could be formulated entirely in terms of block matrices. A centrosymmetric matrix of even order n = 2m can be written in the form