Unitary similarity classes within the cospectral-congruence class of a matrix

• Type: Article in International Scientific Journal
• Date: 2005

Authors

Susana Borges Furtado (Author) FEP; Johnson, CR (Author) Other

Journal

Title: Linear Algebra and Its Applications

Vol. 394

Pages: 291-307

ISSN: 0024-3795

Publisher: Elsevier

Indexing

ISI Web of Knowledge - 3 Citations

Scopus - 0 Citations

Other information

Authenticus ID: P-000-66G

DOI: 10.1016/j.laa.2004.07.019

Abstract (EN): Matrix B epsilon M-n (C) is C-S equivalent (resp. C-E equivalent) to A epsilon M-n (C) if B is both congruent and similar to (resp. cospectral with) A. We are concerned with the number (typically one or infinitely many) of unitary similarity classes in the C-S (resp. C-E) equivalence class of a given matrix. The case n = 2 and the general normal case are fully understood for C-S equivalence. Also, the singular case may generally be reduced to the nonsingular case. The present work includes four main results. (1) If 0 lies in the interior of the field of values of a nonsingular A epsilon M-n, n greater than or equal to 3, then the C-E equivalence class contains infinitely many unitary similarity classes. (2) When 0 is not in the interior, general sufficient conditions are given for the C-E class (and thus the C-S class) to contain only one unitary class. (3) When n = 3, these conditions are also necessary and a classification of all C-E and C-S classes is given. (4) For n greater than or equal to 3, it is shown that the matrices for which the C-S class contains infinitely many unitary similarity classes are dense among all matrices.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 17