Open AccessReal dimension of the Lie algebra of S-skew-Hermitian matrices
Author(s) -
Jonathan Caalim,
Yuichi Tanaka
Publication year - 2022
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/ela.2022.5443
Subject(s) - hermitian matrix , mathematics , invertible matrix , dimension (graph theory) , combinatorics , transpose , lie algebra , matrix (chemical analysis) , pure mathematics , algebra over a field , physics , eigenvalues and eigenvectors , materials science , quantum mechanics , composite material
Let $M_n(\mathbb{C})$ be the set of $n\times n$ matrices over the complex numbers. Let $S \in M_n(\mathbb{C})$. A matrix $A\in M_n(\mathbb{C})$ is said to be $S$-skew-Hermitian if $SA^*=-AS$ where $A^*$ is the conjugate transpose of $A$. The set $\mathfrak{u}_S$ of all $S$-skew-Hermitian matrices is a Lie algebra. In this paper, we give a real dimension formula for $\mathfrak{u}_S$ using the Jordan block decomposition of the cosquare $S(S^*)^{-1}$ of $S$ when $S$ is nonsingular.