“Odd” Matrices and Eigenvalue Accuracy

A definition of even and odd matrices is given, and some of their elementary properties stated. The basic result is that if λ is an eigenvalue of an odd matrix, then so is −λ. Starting from this, there is a consideration of some ways of using odd matrices to test the order of accuracy of eigenvalue routines. 1. Definitions and some elementary properties Let us call a matrix W even if its elements are zero unless the sum of the indices is even – i.e. Wij = 0 unless i + j is even; and let us call a matrix B odd if its elements are zero unless the sum of the indices is odd – i.e. Bij = 0 unless i + j is odd. The non-zero elements of W and B (the letters W and B will always denote here an even and an odd matrix, respectively) can be visualised as lying on the white and black squares, respectively, of a chess board (which has a white square at the top left-hand corner). Obviously, any matrix A can be written as W+B; we term W and B the even and odd parts, respectively, of A. Under multiplication, even and odd matrices have the properties, similar to those of even and odd functions, that even × even and odd × odd are even, even × odd and odd × even are odd. From now on, we consider only square matrices. It is easily seen that, if it exists, the inverse of an even matrix is even, the inverse of an odd matrix is odd. It is also easily seen that in the PLU decomposition of a non-singular matrix A which is either even or odd, L and U are always even, while P is even or odd according as A is. 2010 Mathematics Subject Classification. 15A18,15A99,65F15.