The sum of two phi S orthogonal matrices when S-T S is normal and-1 is not an element of sigma(S-T S)

摘要

Let a nonsingular S is an element of M-n (C) be given. For A is an element of Mn (C), set phi(S) (A) = S(-1)A(T)S. We say that A is phi(S) symmetric if phi(S) (A) = A; we say that A is phi(S) orthogonal if A is an element of GL(n), and phi(S) (A) = A(-1); we say that A has a phi(S) polar decomposition if A = UP for some phi(S) orthogonal U and phi(S) symmetric P. Suppose that S-TS is normal and -1 is not an element of sigma(S-T S). We determine conditions on A is an element of M-n (C) so that A can be written as a sum of two phi(S) orthogonal matrices. (C) 2016 Elsevier Inc. All rights reserved.