The equality of generalized matrix functions on the set of all symmetric matrices

A generalized matrix function d chi(G) : M-n(C) -> C is a function constructed by a subgroup G of S-n and a complex valued function chi of G. The main purpose of this paper is to find a necessary and sufficient condition for the equality of two generalized matrix functions on the set of all symmetric matrices, S-n (C). In order to fulfill the purpose, a symmetric matrix S-sigma is constructed and d(chi)(G) (S-sigma) is evaluated for each sigma is an element of S-n. By applying the value of d(chi)(G) (S-sigma), it is shown that d(chi)(G) (AB) = d(chi)(G)(A)d(chi)(G)(B) for each A, B is an element of S-n (C) if and only if d(chi)(G) = det. Furthermore, a criterion when d(chi)(G)(AB) = d(chi)(G) (BA) for every A, B is an element of S-n (C), is established. (C) 2018 Elsevier Inc. All rights reserved.