Products of two involutions with prescribed eigenvalues and some applications
type: Item
instance of: scholarly article
title: Products of two involutions with prescribed eigenvalues and some applications
zbMATH Open document ID: 1154.15014
author: S. Furtado
published in: Linear Algebra and its Applications
publication date: 2008-10-06T00:00:00Z
review text: Let \(A\) be an \(n\times n\) invertible matrix over a field \(F\). Then \textit{D. Ž. Djoković} [Arch. Math. 18, 582--584 (1967; Zbl 0153.35502)] proved that \(A\) is a product \(BC\) of two involutions (matrices whose square is \(I\)) if and only if \(A\) is similar to \(A^{-1}\). The main theorem of the present paper is the following. If \(\text{char}(F)\neq2\) then the involutions \(B\) and \(C\) can be chosen to have, respectively, \(r\) and \(s\) eigenvalues equal to \(1\) if and only if: (i) \(\det(A)=(-1)^{r+s}\); and (ii) \(\left| s+r-n\right| \leq k_{1}\) and \(\left| s-r\right| \leq k_{2}\) where \(k_{1}\) and \(k_{2}\) are the number of elementary divisors of \(A\) of odd degree associated with the eigenvalues \(1\) and \(-1\), respectively. Using similar methods the author also proves related results such as the following theorem. Suppose that \(C\) is an \(n\times n\) matrix over a field \(F\) where \(F\) is algebraically closed with \(\text{char}(F)\neq2.\) Then \(C\) is similar to a matrix of the form \[ \left[ \begin{matrix} 0 & X\\ Y^{\top} & 0 \end{matrix} \right] \text{ where }X\text{ and }Y\text{ are }s\times(n-s)\text{ matrices} \] if and only if (i) \(C\) and \(-C\) are similar; and (ii) \(\left| 2s-n\right| \leq k\) where \(k\) is the number of elementary divisors of \(C\) of odd degree associated with the eigenvalue \(0\).
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A29
zbMATH DE Number: 5349140
zbMATH Keywords: factorization
zbMATH Keywords: involution
zbMATH Keywords: eigenvalue
zbMATH Keywords: completion
zbMATH Keywords: inverse
MaRDI profile type: Publication
full work available at URL: j.laa.2008.04.044
OpenAlex ID: W1990271529
cites work: Similarity invariants of completions of submatrices
cites work: Product of two involutions
cites work: Q4040931
cites work: Products of real matrices with prescribed characteristic polynomials
cites work: Q3261425
cites work: Matrices with prescribed characteristic polynomial and a prescribed submatrix. III
cites work: Q4055796
cites work: Imbedding conditions for \(\lambda\)-matrices
cites work: Q3203157
cites work: Matrices with prescribed eigenvalues and blocks
cites work: A factorization theorem for matrices
cites work: Factorization of Singular Matrices
cites work: Existenzsätze in der Theorie der Matrizen und lineare Kontrolltheorie
cites work: Matrices with prescribed rows and invariant factors
DOI: 10.1016/J.LAA.2008.04.044
Recommended article: On the similarity of matrices $AB$ and $BA$ over a field
Recommended article: On the number of invariant factors of matrix products
Recommended article: Elementary divisors of s-symmetric matrices
Recommended article: On a unification result by A. R. Sourour concerning commutators and products of involutions
Recommended article: The product of an involution and a skew-involution
Recommended article: Products of diagonalizable matrices
Recommended article: Products of real matrices with prescribed characteristic polynomials
Recommended article: The eigenvalues of the sum of matrices with prescribed invariant polynomials, II
Recommended article: Q3800196
Recommended article: Q3708923
description: scientific article; zbMATH DE number 5349140