Commuting Pairs and Triples of Matrices and Related Varieties
نویسنده
چکیده
In this note, we show that the set of all commuting d-tuples of commuting n × n matrices that are contained is an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber’s theorem on commuting pairs of matrices is a consequence of the irreducibility 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 Mn(F ), and show that it is irreducible of dimension n − n for n ≤ 4, but reducible, of dimension greater than n − n for n ≥ 7.
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