Inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation

Zheng Jian BAI; Raymond H. CHAN

doi:10.1016/j.tcs.2004.01.017

Inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation

Zheng Jian BAI^*
, Raymond H. CHAN

^*Corresponding author for this work

Research output: Journal Publications › Journal Article (refereed) › peer-review

21 Citations (Scopus)

Abstract

In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors {x_i} _i=1^m in Cⁿ and a set of complex numbers {λ_i}_i=1^m, find a centrosymmetric or centroskew matrix C in R^n×n such that {x_i} _i=1^m and {λ_i}_i=1^m are the eigenvectors and eigenvalues of C, respectively. We then consider the best approximation problem for the IEPs that are solvable. More precisely, given an arbitrary matrix B in R^n×n, we find the matrix C which is the solution to the IEP and is closest to B in the Frobenius norm. We show that the best approximation is unique and derive an expression for it.

Original language	English
Pages (from-to)	309-318
Number of pages	10
Journal	Theoretical Computer Science
Volume	315
Issue number	2-3
DOIs	https://doi.org/10.1016/j.tcs.2004.01.017
Publication status	Published - 6 May 2004
Externally published	Yes

Funding

The research was partially supported by the Hong Kong Research Grant Council Grant CUHK4243/01P and CUHK DAG 2060220.

Keywords

Centroskew matrix
Centrosymmetric matrix
Eigenproblem

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10.1016/j.tcs.2004.01.017

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title = "Inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation",

abstract = "In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors \{xi\} i=1m in Cn and a set of complex numbers \{λi\}i=1m, find a centrosymmetric or centroskew matrix C in Rn×n such that \{xi\} i=1m and \{λi\}i=1m are the eigenvectors and eigenvalues of C, respectively. We then consider the best approximation problem for the IEPs that are solvable. More precisely, given an arbitrary matrix B in Rn×n, we find the matrix C which is the solution to the IEP and is closest to B in the Frobenius norm. We show that the best approximation is unique and derive an expression for it.",

keywords = "Centroskew matrix, Centrosymmetric matrix, Eigenproblem",

author = "BAI, \{Zheng Jian\} and CHAN, \{Raymond H.\}",

year = "2004",

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doi = "10.1016/j.tcs.2004.01.017",

language = "English",

volume = "315",

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journal = "Theoretical Computer Science",

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TY - JOUR

T1 - Inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation

AU - BAI, Zheng Jian

AU - CHAN, Raymond H.

PY - 2004/5/6

Y1 - 2004/5/6

N2 - In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors {xi} i=1m in Cn and a set of complex numbers {λi}i=1m, find a centrosymmetric or centroskew matrix C in Rn×n such that {xi} i=1m and {λi}i=1m are the eigenvectors and eigenvalues of C, respectively. We then consider the best approximation problem for the IEPs that are solvable. More precisely, given an arbitrary matrix B in Rn×n, we find the matrix C which is the solution to the IEP and is closest to B in the Frobenius norm. We show that the best approximation is unique and derive an expression for it.

AB - In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors {xi} i=1m in Cn and a set of complex numbers {λi}i=1m, find a centrosymmetric or centroskew matrix C in Rn×n such that {xi} i=1m and {λi}i=1m are the eigenvectors and eigenvalues of C, respectively. We then consider the best approximation problem for the IEPs that are solvable. More precisely, given an arbitrary matrix B in Rn×n, we find the matrix C which is the solution to the IEP and is closest to B in the Frobenius norm. We show that the best approximation is unique and derive an expression for it.

KW - Centroskew matrix

KW - Centrosymmetric matrix

KW - Eigenproblem

UR - https://www.scopus.com/pages/publications/2042511013

U2 - 10.1016/j.tcs.2004.01.017

DO - 10.1016/j.tcs.2004.01.017

M3 - Journal Article (refereed)

AN - SCOPUS:2042511013

SN - 0304-3975

VL - 315

SP - 309

EP - 318

JO - Theoretical Computer Science

JF - Theoretical Computer Science

IS - 2-3

ER -

Inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation

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Keywords

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