# Eigenvalues and eigenvectors of symmetric centrosymmetric matrices

A. Cantoni, P. Butler

Research output: Contribution to journalArticle

224 Citations (Scopus)

### Abstract

It is proved that the eigenvectors of a symmetric centrosymmetric matrix of order N are either symmetric or skew symmetric, and that there are ⌈N/2⌉ symmetric and ⌊N/2⌋ skew symmetric eigenvectors. Some previously known but widely scattered facts about symmetric centrosymmetric matrices are presented for completeness. Special cases are considered, in particular tridiagonal matrices of both odd and even order, for which it is shown that the eigenvectors corresponding to the eigenvalues arranged in descending order are alternately symmetric and skew symmetric provided the eigenvalues are distinct.

Original language English 275-288 14 Linear Algebra and Its Applications 13 3 https://doi.org/10.1016/0024-3795(76)90101-4 Published - 1 Jan 1976 Yes

### Fingerprint

Centrosymmetric Matrix
Eigenvalues and Eigenvectors
Symmetric matrix
Eigenvalues and eigenfunctions
Skew
Eigenvector
Eigenvalue
Tridiagonal matrix
Completeness
Odd
Distinct

### Cite this

@article{25671c8fee5145d39d89d096f5ee182a,
title = "Eigenvalues and eigenvectors of symmetric centrosymmetric matrices",
abstract = "It is proved that the eigenvectors of a symmetric centrosymmetric matrix of order N are either symmetric or skew symmetric, and that there are ⌈N/2⌉ symmetric and ⌊N/2⌋ skew symmetric eigenvectors. Some previously known but widely scattered facts about symmetric centrosymmetric matrices are presented for completeness. Special cases are considered, in particular tridiagonal matrices of both odd and even order, for which it is shown that the eigenvectors corresponding to the eigenvalues arranged in descending order are alternately symmetric and skew symmetric provided the eigenvalues are distinct.",
author = "A. Cantoni and P. Butler",
year = "1976",
month = "1",
day = "1",
doi = "10.1016/0024-3795(76)90101-4",
language = "English",
volume = "13",
pages = "275--288",
journal = "Linear Algebra and Its Applications",
issn = "0024-3795",
publisher = "Elsevier",
number = "3",

}

In: Linear Algebra and Its Applications, Vol. 13, No. 3, 01.01.1976, p. 275-288.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Eigenvalues and eigenvectors of symmetric centrosymmetric matrices

AU - Cantoni, A.

AU - Butler, P.

PY - 1976/1/1

Y1 - 1976/1/1

N2 - It is proved that the eigenvectors of a symmetric centrosymmetric matrix of order N are either symmetric or skew symmetric, and that there are ⌈N/2⌉ symmetric and ⌊N/2⌋ skew symmetric eigenvectors. Some previously known but widely scattered facts about symmetric centrosymmetric matrices are presented for completeness. Special cases are considered, in particular tridiagonal matrices of both odd and even order, for which it is shown that the eigenvectors corresponding to the eigenvalues arranged in descending order are alternately symmetric and skew symmetric provided the eigenvalues are distinct.

AB - It is proved that the eigenvectors of a symmetric centrosymmetric matrix of order N are either symmetric or skew symmetric, and that there are ⌈N/2⌉ symmetric and ⌊N/2⌋ skew symmetric eigenvectors. Some previously known but widely scattered facts about symmetric centrosymmetric matrices are presented for completeness. Special cases are considered, in particular tridiagonal matrices of both odd and even order, for which it is shown that the eigenvectors corresponding to the eigenvalues arranged in descending order are alternately symmetric and skew symmetric provided the eigenvalues are distinct.

UR - http://www.scopus.com/inward/record.url?scp=0016892395&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(76)90101-4

DO - 10.1016/0024-3795(76)90101-4

M3 - Article

VL - 13

SP - 275

EP - 288

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 3

ER -