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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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.

Article information

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

Dates
First available in Project Euclid: 7 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1396833509

Digital Object Identifier
doi:10.15352/afa/1396833509

Mathematical Reviews number (MathSciNet)
MR3192016

Zentralblatt MATH identifier
1297.15010

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

Keywords
The problem of A. Horn normal matrices upper Hessenberg

Citation

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. https://projecteuclid.org/euclid.afa/1396833509


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References

  • T. Ando and R. Bhatia, Eigenvalue inequalities associated with the Cartesian decomposition, Linear Multilinear Algebra 22 (1987), no. 2, 133–147.
  • Lei Cao, A new formulation and uniqueness of solutions to A. Horn's problem, Dissertation, 2012.
  • G.H. Golub and C.F. Van Loan. Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, fourth edition, 2013.
  • H. Hochstadt, On the construction of a Jacobi matrix from spectral data, Linear Algebra Appl. 8 (1974), 435–446.
  • A. Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225–241.
  • A.A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.) 4 (1998), no. 3, 419–445.
  • A. Knutson and T. Tao, The honeycomb model of ${\rm GL}_n({\bf C})$ tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055–1090.
  • A. Knutson, T. Tao and C. Woodward, The honeycomb model of ${\rm GL}_n(\Bbb C)$ tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc. 17 (2004), no. 1, 19–48 (electronic).
  • C.-K. Li, Y.-T. Poon and N.-S. Sze, Eigenvalues of the sum of matrices from unitary similarity orbits, SIAM J. Matrix Anal. Appl. 30 (2008), no. 2, 560–581.
  • S.M. Malamud, Inverse spectral problem for normal matrices and the Gauss-Lucas theorem, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4043–4064.
  • H. Wielandt, On eigenvalues of sums of normal matrices, Pacific J. Math. 5 (1955), 633–638.