The contragredient equivalence for several matrices: A set of invariants
type: Item
instance of: scholarly article
title: The contragredient equivalence for several matrices: A set of invariants
zbMATH Open document ID: 0944.15009
DOI: 10.1016/S0024-3795(98)10212-4
author: V. Pereyra
published in: Linear Algebra and its Applications
publication date: 2000-09-18T00:00:00Z
review text: An \(n\)-tuple of complex matrices \((A_1,A_2,\dots,A_n)\) is said to be cyclically multiplicable if the products \(A_iA_{i+1}\) are defined for all \(i\) modulo \(n\). Two cyclically multiplicable \(n\)-tuples of matrices \((A_1,A_2,\dots,A_n)\) and \((A_1',A_2',\dots,A_n')\) are contragrediently equivalent if there exist nonsingular matrices \(T_1,T_2,\dots,T_n\) such that \(A_i'=T_i^{-1}A_iT_{i+1}\) for all \(i\) modulo \(n\). When this holds, clearly the square matrices \(B_i=A_iA_{i+1}\dots A_{i-1}\) and \(B_i'=A_i'A_{i+1}'\dots A_{i-1}'\) are similar for all \(i\) modulo \(n\). The main problem studied and solved in a natural but quite technical manner in this well written paper is to find a necessary and sufficient condition for two cyclically multiplicable \(n\)-tuples to be contragrediently equivalent. The condition is that they have the same ``spectra of divisors''. Unfortunately a few pages of technical definitions and preliminary results are needed to fully explain the result. This work builds on substantial earlier work by several authors (including the present one), beginning with the fundamental paper by \textit{H. Flanders}, [Proc. Am. Math. Soc. 2, 871-874 (1951; Zbl 0044.00602)].
reviewed by: Stanley E. Payne
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 1331314
zbMATH Keywords: cyclically multiplicable matrices
zbMATH Keywords: contragredient equivalence
zbMATH Keywords: elementary divisors
MaRDI profile type: Publication
cites work: Elementary Divisors of AB and BA
cites work: An extension of flanders theorem to several matrices
cites work: Contragredient equivalence: A canonical form and some applications
cites work: The Relationship Between AB and BA
cites work: Q4835997
Recommended article: Spectral variation under congruence
Recommended article: Spectral variation under congruence for a nonsingular matrix with 0 on the boundary of its field of values
Recommended article: Cospectrality and similarity for a pair of matrices under multiplicative and additive composition with diagonal matrices
Recommended article: On similarity of matrices over commutative rings
Recommended article: Similarity transformations of decomposable matrix polynomials and related questions
Recommended article: On the similarity of matrices $AB$ and $BA$ over a field
Recommended article: Concomitants etp-uplets de matrices 2 × 2
Recommended article: On divisibility and factorization of matrices.
Recommended article: Q4214910
Recommended article: Order invariant spectral properties for several matrices
description: scientific article; zbMATH DE number 1331314