The determinant of the sum of two normal matrices with prescribed eigenvalues
This paper presents some partial results on a conjecture of the present reviewer: let \(A\) and \(B\) be two \(n\times n\) normal matrices with eigenvalues \(\alpha_ 1,\ldots,\alpha_ n\) and \(\beta_ 1,\ldots,\beta_ n\), respectively. Then the determinant of \(A+B\) belongs to the convex hull of the points \(\prod_{i}(\alpha_ i+\beta_{\sigma}+\beta_{\sigma (i)})\), \(\sigma\) running over the symmetric group \(S_ n\) of degree \(n\). The deepest result is perhaps the following. Let \(\Delta\) be the set described by the determinant of \(A+B\) when \(A\) and \(B\) run over all the normal matrices which have the above eigenvalues and denote by \(\partial \Delta\) its boundary. If \(z\) belongs to \(\partial \Delta\) and is different from zero and if in a neighborhood of \(z\), \(\Delta\) is contained in an angle with vertex at \(z\) and measuring less than \(\pi\), then there is a \(\sigma\) in \(S_ n\) such that \[ z=\prod_{i}(\alpha_ j+\beta_{\sigma (j)}). \]
scientific article; zbMATH DE number 3920681