Block similarity invariants of restrictions to \((A,B)\)-invariant subspaces
type: Item
instance of: scholarly article
title: Block similarity invariants of restrictions to \((A,B)\)-invariant subspaces
zbMATH Open document ID: 0824.15012
DOI: 10.1016/0024-3795(94)00127-Y
published in: Linear Algebra and its Applications
publication date: 1995-07-03T00:00:00Z
review text: Let be given a pair of matrices \((A, B)\) over a commutative field \(F\), where \(A\) is \(p\times p\) and \(B\) is \(q\times p\). Block similarity invariants of \((A, B)\) can be identified with the Kronecker invariants of the singular pencil \[ \lambda\biggl[\begin{smallmatrix} I_ p\\ 0\end{smallmatrix}\biggr]- \biggl[\begin{smallmatrix} A\\ B\end{smallmatrix}\biggr], \] i.e., the row-minimal indices and the invariant factors of the pencil. A complete characterization is obtained (in terms of divisibility relations) of the possible block similarity invariants of pairs of matrices of the form \[ \biggl(\biggl[\begin{smallmatrix} A_ 1\\ 0\end{smallmatrix} \begin{smallmatrix} A_ 3\\ A_ 2\end{smallmatrix}\biggr],\;\biggl[\begin{smallmatrix} B_ 1\\ 0\end{smallmatrix} \begin{smallmatrix} B_ 3\\ B_ 2\end{smallmatrix}\biggr]\biggr), \] where \((A_ 1, B_ 1)\) is a pair of matrices with given block similarity invariants. The proof is based on interlacing inequalities for invariant factors of polynomial matrices, canonical form of matrix pairs under block similarity, and the well-known connections between the invariant polynomials of a linear transformation in a finite-dimensional vector space and those of the restriction of the linear transformation to its invariant subspace. The main result is also presented in a basis free formulation, using the notion of invariant subspaces for linear transformations between two different finite-dimensional vector spaces.
reviewed by: Leiba Rodman
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 769018
zbMATH Keywords: Kronecker invariants
zbMATH Keywords: singular pencil
zbMATH Keywords: invariant factors
zbMATH Keywords: block similarity invariants
zbMATH Keywords: interlacing inequalities
zbMATH Keywords: polynomial matrices
zbMATH Keywords: canonical form
zbMATH Keywords: matrix pairs
zbMATH Keywords: invariant subspace
author: Itziar Baragaña
author: Ion Zaballa
MaRDI profile type: Publication
cites work: Column completion of a pair of matrices
cites work: Q5597890
cites work: Q5580324
cites work: Similarity of non-everywhere defined linear maps
cites work: Q4097043
cites work: Similarity of operator blocks and canonical forms. I. General results, feedback equivalence and Kronecker indices
cites work: Q3748420
cites work: Q5525419
cites work: Imbedding conditions for \(\lambda\)-matrices
cites work: Interlacing inequalities for invariant factors
description: scientific article; zbMATH DE number 769018