A normal variation of the Horn problem: the rank 1 case

Abstract

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 $\lambda_k=0,k=2,\ldots,n,$ 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.