Solution of the congruence problem for arbitrary hermitian and skew-hermitian matrices over polynomial rings
D. Z. Djokovic, F. Szechtman
Abstract
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.
Create a lesson
Related papers
Rota-Baxter operators and factorization of Lie affgebras
JiaJia Han, Abror Khudoyberdiyev, Yunzhou Xie et al.
Diagonalization of skew-Hermitian dual quaternion matrices and ϕ-Hermitian dual quaternion matrices
Cailing Yao, Bingzhe Hou
The E6 normal forms: semisimple deformations, geometry, DT invariants, and the E6,∞ order
Shengyong Pan
A 9-dimensional nilpotent Lie algebra isomorphic to its complex conjugate admitting no real form
Tom De Medts
Projective Differential Moment Invariants from the Viewpoint of Classical Invariant Theory
Leonid Bedratyuk
Global obstruction property and its applications
Weiheng Xia