On the elementary symmetric functions of a sum of matrices

TL;DR

This paper explores the elementary symmetric functions of a sum of matrices, providing new formulas, connections with the Möbius function and posets, and solving a related determinant problem.

Contribution

It introduces novel expressions for the determinant of matrix sums and links with combinatorial structures like Möbius functions and posets.

Findings

New formulas for elementary symmetric functions of matrix sums

Connection established with Möbius functions and posets

Solved a specific problem related to the determinant of sum of matrices

Abstract

Often in mathematics it is useful to summarize a multivariate phenomenon with a single number and in fact, the determinant -- which is represented by det -- is one of the simplest cases. In fact, this number it is defined only for square matrices and a lot of its properties are very well-known. For instance, the determinant is a multiplicative function, i.e. det(AB)=detA detB, but it is not, in general, an additive function. Another interesting function in the matrix analysis is the characteristic polynomial -- in fact, given a matrix A, this function is defined by $p_A(t)=det(tI-A)$ where I is the identity matrix -- which elements are, up a sign, the elementary symmetric functions associated to the eigenvalues of the matrix A. In the present paper new expressions related with the determinant of sum of matrices and the elementary symmetric functions are given. Moreover, the connection with the Mobius function and the partial ordered sets (poset) is presented. Finally, a problem related with the determinant of sum of matrices is solved.