Description:
<jats:p>The purpose of this paper is to study generalized matrix functions only
using the permutation matrices and symmetric matrices. Firstly the zeroness
of a generalized matrix function and then the equality of two generalized
matrix functions on the permutation matrices and symmetric matrices will be
examined. Secondly generalized matrix functions preserving commutativity of
the permutation matrices or commutativity of the symmetric matrices will be
characterized. Thirdly generalized matrix functions which preserve product
of the permutation matrices or product of the symmetric matrices will be
investigated. Finally the Cayley-Hamilton Theorem for generalized
characteristic polynomials using the permutation matrices and symmetric
matrices will be studied.</jats:p>