A family of symmetric polynomials of the eigenvalues of a matrix

In this paper, we consider a family of symmetric polynomials of the eigenvalues of a complex matrix A and find an explicit expression of each member of the family as a polynomial of the entries of A with positive coefficients. In the case of a nonnegative matrix, one immediately obtains a family of inequalities involving matrix eigenvalues and diagonal entries. Equivalent forms of some of the obtained results as well as connections with known results and specific applications are also presented. In the concluding part of the paper, we provide comments and conjecture further inequalities related with nonnegative matrices.