Linear algebra in several variables

Abstract: Many mathematical and scientific problems concern systems of linear operators $(A_1, ..., A_n)$. Spectral theory is expected to provide a mechanism for studying their properties, just like the case for an individual operator. However, defining a spectrum for non-commuting operator systems has been a difficult task. The challenge stems from an inherent problem in finite dimension: is there an analogue of eigenvalues in several variables? Or equivalently, is there a suitable notion of joint characteristic polynomial for multiple matrices $A_1, ..., A_n$? A positive answer to this question seems to have emerged in recent years.

Definition. Given square matrices $A_1, ..., A_n$ of equal size, their characteristic polynomial is defined as \[Q_A(z):=\det(z_0I+z_1A_1+\cdots+z_nA_n), z=(z_0, ..., z_n)\in \mathbb{C}^{n+1}.\] Hence, a multivariable analogue of the set of eigenvalues is the eigensurface (or eigenvariety) $Z(Q_A):=\{z\in \mathbb{C}^{n+1}\mid Q_A(z)=0\}$. This talk will review some applications of this idea to problems involving projection matrices and finite dimensional complex algebras. The talk is self-contained and friendly to graduate students.

combinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

OIST representation theory seminar

Series comments: Timings of this seminar may vary from week to week.