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Matrix Behaviour, Unitary Reducibility, and Hadamard Products

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Abstract

The question investigated here is: if two matrices A and B in \CNN have identical behaviour in a unitarily invariant norm \norm, \ie,\ \normp(A)=\normp(B) for every polynomial p with complex coefficients, what properties do A and B have in common? After a preliminary result about eigenvalues, it is shown with a mildly restrictive assumption that if A is unitarily reducible, so is B. A theorem is proved about Hadamard products of the form H∘\invtH, where H is Hermitian positive definite. Finally, an example is produced where A and B have identical behaviour in the Frobenius norm, but are not related to each other in any simple way.

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1996-07

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Cornell University

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computer science; technical report

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http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR96-1596

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technical report

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