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Super‐identical pseudospectra
Author(s) -
Bourque Maxime Fortier,
Ransford Thomas
Publication year - 2009
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn085
Subject(s) - integer (computer science) , mathematics , invariant (physics) , combinatorics , pure mathematics , complex matrix , mathematical physics , computer science , chemistry , chromatography , programming language
The complex N × N matrices A and B are said to have super‐identical pseudospectra if, for each z ∈ ℂ, the singular values of A − zI are the same as those of B − zI . We explore this condition and its consequences. On the positive side, drawing on ideas from invariant theory, we prove that there exists an integer m = m ( N ) such that ‘almost every’ m ‐tuple of N × N matrices with super‐identical pseudospectra contains a pair that are unitarily equivalent. On the negative side, we present an example of a pair of non‐derogatory 4 × 4 matrices A and B with super‐identical pseudospectra such that || A 2 || ≠ || B 2 ||.