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

On the elementary symmetric functions of a sum of matrices

Costas-Santos, R. S. JP Journal of Algebra, Number Theory: Advances and Applications 1, No. 2 (2009), 99 — 112

Pushpa Publishing House (INDIA) | ISSN: 0972-5555 | SCOPUS®SCImago Journal Rank (SJR) Subject Category: Algebra and Number Theory -- position: n/a


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 · det B, A, B ∈ Mn, and it is a multilinear function, but it is not, in general, an additive function, i.e. det(A + B) is not equal to det A + det B.

Another interesting scalar function in the Matrix Analysis is the characteristic polyno- mial. In fact, given a square matrix A, the coefficients of its characteristic polynomial χA(t) := det(tI − 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 func- tions 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 + tB)m) ∈ R[t], has only nonnegative coefficients whenever A, B ∈ Mr 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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@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},
YEAR = {2009},
PAGES = {99--112},
ISSN = {0975-1548},
ZBL = {1275.15005},
MRCLASS = {11C20 (05E05 11P81)},