Generalized matrix functions, determinant and permanent

چکیده مقاله

Since linear and multilinear algebra has many applications in different
branches of sciences, the attention of many mathematicians has been attracted
to it in recent decades. The determinant and the permanent are the most
important functions in linear algebra and so a generalized matrix function,
which is a generalization of the determinant and the permanent, becomes
significant. Generalized matrix functions connect some branches of
mathematics such as theory of finite groups, representation theory of groups,
graph theory and combinatorics, and linear and multilinear algebra.

In this paper, using permutation matrices or symmetric matrices, necessary
and sufficient conditions are given for a generalized matrix function to be the
determinant or the permanent. We prove that a generalized matrix function is
the determinant or the permanent if and only if it preserves the product of
symmetric permutation matrices. Also we show that a generalized matrix
function is the determinant if and only if it preserves the product of symmetric
matrices.