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

Abstract

A generalized matrix function d G χ : M n (C) → C is a function constructed by a subgroup G of S n and a complex valued function χ 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 σ is constructed and d G χ (S σ ) is evaluated for each σ ∈ S n . By applying the value of d G χ (S σ ), it is shown that d G χ (AB) = d G χ (A)d G χ (B) for each A, B ∈ S n (C) if and only if d G χ = det. Furthermore, a criterion when d G χ (AB) = d G χ (BA) for every A, B ∈ S n (C), is established.

Generalized matrix functions on a linear sum of permutation matrices and their cousins

“…In [6], we introduced an n × n symmetric matrix S θ , where θ ∈ S n . The formula for a generalized matrix function of S θ were also investigated.…”

“…A concept of these matrices can be extended to generalized permutation matrices. In [6], the authors worked with symmetric matrices constructed by permutations. These matrices can be viewed as a sum of two generalized permutation matrices, denoted by S σ .…”