Commuting pairs and triples of matrices and related varieties

In this note, we show that the set of all commuting d-tuples of commuting n×n matrices that are contained in an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber's theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuti...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 310; no. 1; pp. 139 - 148
Main Authors Guralnick, Robert M., Sethuraman, B.A.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.05.2000
Subjects
Commutative subalgebras of matrix algebras
Commuting matrices
Commuting variety
13A50
Commutative subalgebras of matrix algebras
15A72
13D40
16R30
Commuting matrices
Commuting variety
16G20
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Summary:In this note, we show that the set of all commuting d-tuples of commuting n×n matrices that are contained in an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber's theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also study the variety of n-dimensional commutative subalgebras of M n(F) , and show that it is irreducible of dimension n 2−n for n⩽4, but reducible, of dimension greater than n 2−n for n⩾7.
ISSN:0024-3795
1873-1856
DOI:10.1016/S0024-3795(00)00065-3