rscosan.com

Talk: On the determinant of a sum of matrices

On the determinant of a sum of matrices

Date: 2008..10..18
Event: RCTM08. Conference in honor of Robert C. Thompson
Venue: Universidad de California, Santa Barbara. CA, USA

Abstract

Often in mathematics it is useful to summarize a multivariate phenomenon with a single number. In fact, the determinant — which is denoted by det — is one of the simplest cases and many of its properties are very well-known. For instance, the determinant is a multiplicative function, i.e. $$\det(A B) = \det A \cdot \det B$$, $$A, B \in M_n$$, and it is a multilinear function, but it is not, in general, an additive function, i.e. $$\det(A + B) \not = \det A + \det B$$.

Another interesting scalar function in the Matrix Analysis is the characteristic polynomial. In fact, given a square matrix A, the coefficients of its characteristic polynomial $$\chi_A(t):=\det(t I-A)$$ are, up to a sign, the elementary symmetric functions associated with the eigenvalues of $$A$$.

In this talk we present new expressions related to the elementary symmetric functions of sum of matrices.