A family of symmetric polynomials of the eigenvalues of a matrix
Let \(A\) be an \(n\times n\) matrix, and denote by \(\Lambda=(\lambda_1,\dots,\lambda_n)\) its spectrum. For \(k=0,1,2,\dots\) and \(1\leq m\leq n\), consider the symmetric polynomial of the eigenvalues defined by \[ s_{k,m}(\Lambda):=\sum_{{{1\leq i_1<\dots<i_m\leq n}}\atop{{j_1+\dots +j_m=k}} }\lambda_{i_1}^{j_1}\lambda_{i_2}^{j_2}\cdots \lambda_{i_m}^{j_m}. \] The main theorem expresses \(s_{k,m}(\Lambda)\) as a homogeneous polynomial of degree \(k\) in the entries of \(A\). In particular, this leads to an upper bound for \(s_{k,m}(\Lambda)\) in terms of the diagonal entries of \(A\) in the case where \(A\) is positive. Another application is a formula relating \(s_k\), the sum of the \(k\)th powers of the roots of a polynomial \(p\), to \(s_k'\), the corresponding sum for the derived polynomial \(p'\). The paper concludes with the conjecture that if \(p\) is the characteristic polynomial of a nonnegative matrix \(A\), then these sums \(s_k'\) are nonnegative (see [the author, Cybern. Inform. Technol. 2, 3--11 (2006)], for the motivation to this conjecture).
scientific article; zbMATH DE number 5349182