Paper: On the elementary symmetric functions of a sum of matrices

On the elementary symmetric functions of a sum of matrices

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 the present paper we present new expressions related to the elementary symmetric functions of sum of matrices. The main motivation of this manuscript is try to find new properties to probe the following conjecture:

Bessis-Moussa-Villani conjecture: The polynomial \(p(t) :=\)Tr\(((A+B\, t)^m) \in \mathbb R[t]\), has only nonnegative coefficients whenever \(A, B \in M_r\) are positive semidefinite matrices.

Moreover, some numerical evidences and the Newton-Girard formulas suggested to us to consider a more general conjecture that will be considered in a further manuscript.

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BibTeX

@article {cos1,
AUTHOR={Costas-Santos, Roberto S.},
TITLE={On the elementary symmetric functions of a sum of matrices},
JOURNAL={J. Algebra Number Theory, Adv. Appl.},
FJOURNAL={Journal of Algebra, Number Theory: Advances and Applications},
VOLUME={1},
NUMBER={2},
YEAR={2009},
PAGES={99--112},
ISSN={0975-1548},
ZBL={1275.15005},
MRCLASS={11C20 (05E05 11P81)},
}