Solution of the congruence problem for arbitrary hermitian and skew-hermitian matrices over polynomial rings

Details Solution of the congruence problem for arbitrary hermitian and skew-hermitian matrices over polynomial rings Solution of the congruence problem for arbitrary hermitian and skew-hermitian matrices over polynomial rings

2002-06-10

arxiv.org/abs/math/0206104v1

Mathematical Research Letters 10 (2003), 1-10

Mathematics

Rings and Algebras

9 pages

Scientific paper

We equip the complex polynomial algebra C[t] with the involution which is the identity on C and sends t to -t. Answering a question raised by V.G. Kac, we show that every hermitian or skew-hermitian matrix over this algebra is congruent to the direct sum of 1 by 1 matrices and 2 by 2 matrices with zero diagonal. Morevover, we show that if two n by n hermitian or skew-hermitian matrices have the same invariant factors, then they are congruent. The complex field can be replaced by any algebraically closed field of characteristic not 2.

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