Journal of Algebra, Number Theory: Advances and ApplicationsAuthor's: R. S. Costas-Santos
Pages: [99] - [112]
Received Date: June 11, 2009
Submitted by:
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., 
and it is a multilinear function, but it is
not, in general, an additive function, i.e., 
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 
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: [2, 4]
The polynomial 
has only nonnegative coefficients whenever
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.
Positivity Conjecture :
The polynomial 
has only nonnegative coefficients whenever
are positive semidefinite matrices for every
It is clear that the BMV conjecture is a particular case of the
positivity conjecture for 
since 
elementary symmetric function, Hermitian matrix, determinant.
