Optimization problems on the rank of the solution to left and rightinverse eigenvalue problem

Li-Fang Dai; Mao-Lin Liang; Wei-Yuan Ma

doi:10.3934/jimo.2015.11.171

Article Contents

2015, Volume 11, Issue 1: 171-183. Doi: 10.3934/jimo.2015.11.171

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Optimization problems on the rank of the solution to left and right inverse eigenvalue problem

1.
School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001
2.
School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, Gansu 730030

Received: December 31, 2012

Revised: November 30, 2013

Published: December 2014

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Abstract

A complex matrix $P$ is called Hermitian and $\{k+1\}$-potent if $P^{k+1}=P=P^*$ for some integer $k\geq 1$. Let $P$ and $Q$ be $n\times n$ Hermitian and $\{k+1\}$-potent matrices, we say that complex matrix $A$ is $\{P,Q,k+1\}$-reflexive (anti-reflexive) if $PAQ=A$ ($PAQ=-A$). In this paper, the solvability conditions and the general solutions of the left and right inverse eigenvalue problem for $\{P,Q,k+1\}$-reflexive and anti-reflexive matrices are derived, and the minimal and maximal rank solutions are given. Moreover, the associated optimal approximation problem is also considered. Finally, numerical example is given to illustrate the main results.

Keywords:

Inverse eigenvalue problem,
left and right eigenpairs,
$\{k+1\}$-potent matrix,
$\{P,
Q,
k+1\}$-reflexive (anti-reflexive) matrix,
rank,
optimal approximation.

Mathematics Subject Classification: Primary: 65F18, 15A09; Secondary: 15B99.

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References

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