Moving zeros among matrices

We investigate the zero-patterns that can be created by unitary similarity in a given matrix, and the zero-patterns that can be created by simultaneous unitary similarity in a given sequence of matrices. The latter framework allows a “simultaneous Hessenberg” formulation of Pati’s tridiagonal result...

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Bibliographic Details
Published in:Linear algebra and its applications 2007-07, Vol.424 (1), p.83-95
Main Authors: Holbrook, John, Schoch, Jean-Pierre
Format: Article
Language:English
Subjects:
Algebra
Algebraically closed fields
Exact sciences and technology
Hessenberg forms
Linear and multilinear algebra, matrix theory
Mathematics
Sciences and techniques of general use
Tridiagonal matrices
Unitary similarity
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Summary:We investigate the zero-patterns that can be created by unitary similarity in a given matrix, and the zero-patterns that can be created by simultaneous unitary similarity in a given sequence of matrices. The latter framework allows a “simultaneous Hessenberg” formulation of Pati’s tridiagonal result for 4 × 4 matrices. This formulation appears to be a strengthening of Pati’s theorem. Our work depends at several points on the simplified proof of Pati’s result by Davidson and Djoković. The Hessenberg approach allows us to work with ordinary similarity and suggests an extension from the complex to arbitrary algebraically closed fields. This extension is achieved and related results for 5 × 5 and larger matrices are formulated and proved.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2006.04.010