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Spaces of matrices with a sole eigenvalue

By Clément de Seguins Pazzis

Abstract

Let K be an arbitrary (commutative) field and L be an algebraic closure of it. Let V be a linear subspace of M_n(K), with n>2. We show that if every matrix of V has at most one eigenvalue in K, then dim V<=1+n(n-1)/2. If every matrix of V has a sole eigenvalue in L and dim V=1+n(n-1)/2, we show that V is similar to the space of all upper-triangular matrices with equal diagonal entries, except if n=3 and K has characteristic 3, or if n=4 and K has characteristic 2. In both of those special cases, we classify the exceptional solutions up to similarity.Comment: 33 page

Topics: Mathematics - Rings and Algebras, Primary: 15A30. Secondary: 15A18
Year: 2012
DOI identifier: 10.1080/03081087.2011.654118
OAI identifier: oai:arXiv.org:1105.1872
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