A normal variation of the Horn problem: the rank 1 case. (English) Zbl 1297.15010

Cao, Lei; Woerdeman, Hugo J.

Ann. Funct. Anal. 5, No. 2, 138-146 (2014).

Summary: Given three \(n\)-tuples \(\{\lambda_i\}_{i=1}^n\) , \(\{\mu_i\}_{i=1}^n\), \(\{\nu_i\}_{i=1}^n\) of complex numbers, we introduce the problem of when there exists a pair of normal matrices \(A\) and \(B\) such that \(\sigma (A) = \{\lambda_i\}_{i=1}^n\), \(\sigma (B) = \{\mu_i\}_{i=1}^n\), and \(\sigma (A + B) = \{\nu_i\}_{i=1}^n\), where \(\sigma (\cdot )\) denote the spectrum. In the case when \(\lambda_k = 0\), \(k = 2,\dots , n\), we provide necessary and sufficient conditions for the existence of \(A\) and \(B\). In addition, we show that the solution pair \((A, B)\) is unique up to unitary similarity. The necessary and sufficient conditions reduce to the classical A. Horn inequalities when the \(n\)-tuples are real.

MSC:

15A18	Eigenvalues, singular values, and eigenvectors
15A29	Inverse problems in linear algebra
15B57	Hermitian, skew-Hermitian, and related matrices
15A42	Inequalities involving eigenvalues and eigenvectors

Keywords:

inverse eigenvalue problem; problem of Horn; normal matrices; upper Hessenberg matrix; spectrum; classical Horn inequalities