Annals of Functional Analysis

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

Lei Cao and Hugo J. Woerdeman

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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.

Article information

Ann. Funct. Anal., Volume 5, Number 2 (2014), 138-146.

First available in Project Euclid: 7 April 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A18: Eigenvalues, singular values, and eigenvectors
Secondary: 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)

The problem of A. Horn normal matrices upper Hessenberg


Cao, Lei; Woerdeman, Hugo J. A normal variation of the Horn problem: the rank 1 case. Ann. Funct. Anal. 5 (2014), no. 2, 138--146. doi:10.15352/afa/1396833509.

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