×

Essentially Hermitian matrices revisited. (English) Zbl 1151.15301

Summary: The following case of the determinantal conjecture of M. Marcus [Indiana Univ. Math. J. 22, 1137–1149 (1973; Zbl 0243.15025)] and G. N. de Oliveira [Research problem: Normal matrices. Linear and Multilinear Algebra 12, 153–154 (1982)] is established. Let \(A\) and \(C\) be Hermitian \(n\times n\) matrices with prescribed eigenvalues \(a_1,\dots,a_n\) and \(c_1,\dots,c_n\), respectively. Let \(k\) be a non-real unimodular complex number, \(B=\kappa C\), \(b_j = \kappa c_j\) for \(j=1,\dots,n\). Then
\[ \det(A-B)\in \text{co}\left\{\prod_{j=1}^n (a_j-b_{\sigma(j)});\;\sigma \in S_n\right\}, \]
where \(S_n\) denotes the group of all permutations of \(\{1,\dots,n\}\) and co the convex hull taken in the complex plane.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15B57 Hermitian, skew-Hermitian, and related matrices

Citations:

Zbl 0243.15025
PDFBibTeX XMLCite
Full Text: DOI EuDML Link Link