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.