Many mathematical and scientific problems concern systems of linear operators (A1,…,An). 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 A1,…,An? A positive answer to this question seems to have emerged in recent years.
Definition. Given square matrices A1,…,An of equal size, their characteristic polynomial is defined as
QA(z):=det(z0I + z1A1 + ⋯ + znAn), z=(z0,…,zn) ∈ ℂn+1.
Hence, a multivariable analogue of the set of eigenvalues is the eigensurface (or eigenvariety) Z(QA):={z ∈ ℂn+1 ∣ QA(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.
This video was produced by the Okinawa Institute of Science and Technology, as part of their OIST Representation Theory Seminar series.
