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# Mathematical Research Letters

## Volume 10 (2003)

### Number 1

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

Pages: 1 – 10

DOI: http://dx.doi.org/10.4310/MRL.2003.v10.n1.a1

#### Authors

#### Abstract

Let ${}^*$ be the involutorial automorphism of the complex polynomial algebra ${\mbox{\bf C}}[t]$ which 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 $1\times1$ matrices and $2\times2$ matrices with zero diagonal. Moreover we show that if two $n\times n$ 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 $\ne2$.