分野別セミナー

The non-Hermitian matrix-valued Brownian motion is the stochastic
process of a random matrix whose entries are given by independent
complex Brownian motions. The eigenvector-overlap process is a Hermitian
matrix-valued process, each element of which is given by a product of an
overlap of right eigenvectors and that of left eigenvectors, and the
square roots of its diagonal elements are known as the condition numbers
of eigenvalues in perturbation theory. We derive a set of SDEs for the
coupled system of the eigenvalue process and the eigenvector-overlap
process and discuss the results from the viewpoint of condition numbers.
The regularized Fuglede--Kadison (FK) determinant is introduced which
evolves following a stochastic PDE on the two-dimensional complex space.
Time-dependent point process of eigenvalues and its variation marked by
the squares of the condition numbers are related to the logarithmic
derivatives of the FK-determinant random-field. We also discuss the
pseudospectrum process in the case that the initial matrix is nonnormal.
The present talk is based on the joint works with Syota Esaki (Fukuoka)
and Satoshi Yabuoku (Kitakyushu) (https://arxiv.org/abs/2306.00300) and
with Saori Morimoto (Chuo) and Tomoyuki Shirai (Kyushu).