A bound for the determinant of certain Hadamard products and for the determinant of the sum of two normal matrices
Let \(A\) and \(U\) be complex \(n \times n\) matrices and suppose that either \(\sum^ n_{i = 1} | u_{ij} | \leq 1\) for all \(j\), or \(\sum^ n_{j = 1} | u_{ij} | \leq 1\) for all \(i\). The author proves that \(| \text{det} (A \circ U) | < \max \sum_{\sigma \in S_ n} | a_{i \sigma (i)} |\). In fact, he gives two proofs. Scale \(A\) so that \(\max \sum_{\sigma \in S_ n} | a_{i \sigma (i)} | = 1\). In the first proof he shows that \(| \text{det} (A \circ U) | \leq | \text{det} (Z \circ U) |\) for another matrix \(Z\) with each \(| z_{ij} | = 1\) and then he applies the Hadamard determinant inequality to \(| \text{det} (Z \circ U) |\). In the second proof he uses quite different techniques to show there exists \(B\) with \(| b_{ij} | \leq 1\) and \(| \text{det} (A \circ U) | = | \text{det} (B \circ U) |\), and then he applies the Hadamard inequality. An immediate corollary is: when \(M\) and \(N\) are \(n \times n\) and normal, with eigenvalues \(\{\mu_ i\}\) and \(\{\nu_ i\}\) respectively, then \(| \text{det} (M + N) | \leq \max_{\sigma \in S_ n} | \mu_ i + \nu_{\sigma (i)} |\). This is the best possible upper bound one could expect for \(| \text{det} (M + N) |\) for general normal \(M\) and \(N\), and it generalizes an inequality of \textit{J. F. Queiró} and \textit{A. Kovaček} [Linear Multilinear Algebra 33, No. 3/4, 171-173 (1993; Zbl 0766.15023)]. It also supports the determinantal conjecture of \textit{M. Marcus} [Indiana Univ. Math. J. 22, 1137-1149 (1973; Zbl 0255.15020)] and \textit{G. N. Oliveira} [Research problem: Normal matrices. Linear Multilinear Algebra 12, 153-154 (1982)].
scientific article; zbMATH DE number 566105
scientific article