Diagonal norm hermitian matrices |
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Authors: | Hans Schneider REL Turner |
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Institution: | University of Wisconsin-Madison Madison, Wisconsin, USA |
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Abstract: | If v is a norm on n, let H(v) denote the set of all norm-Hermitians in nn. Let S be a subset of the set of real diagonal matrices D. Then there exists a norm v such that S=H(v) (or S = H(v)∩D) if and only if S contains the identity and S is a subspace of D with a basis consisting of rational vectors. As a corollary, it is shown that, for a diagonable matrix h with distinct eigenvalues λ1,…, λr, r?n, there is a norm v such that h ∈ H(v), but hs?H(v), for some integer s, if and only if λ2–λ1,…, λr–λ1 are linearly dependent over the rationals. It is also shown that the set of all norms v, for which H(v) consists of all real multiples of the identity, is an open, dense subset, in a natural metric, of the set of all norms. |
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