Speaker
Description
Linear preserver problems study linear maps on matrix spaces that leave certain functions, subsets, or relations invariant, whereas matrix decomposition problems focus on expressing matrices as products of matrices with prescribed structural properties. An element of the algebra $M_n(\mathbb{F})$ of $n \times n$ matrices over a field $\mathbb{F}$ is called an involution if its square equals the identity matrix. A classical result of Gustafson, Halmos, and Radjavi asserts that any product of involutions in $M_n(\mathbb{F})$ can be written as a product of at most four involutions. It is also known that a matrix is a product of two involutions if and only if it is similar to its inverse. These results naturally lead to the following linear preserver question: which linear maps on $M_n(\mathbb{F})$ preserve products of involutions?
In this talk, we address this question and present a characterization of bijective linear maps that preserve matrices expressible as products of two, three, or four involutions in $M_n(\mathbb{F})$. We also outline possible directions for further research. This is joint work with Chi-Kwong Li and Sushil Singla.