Smaller Gershgorin disks for multiple eigenvalues for complex matrices

TL;DR

This paper extends Gershgorin disk theory to complex matrices, providing tighter bounds for multiple eigenvalues and applications to normal and almost symmetric matrices, improving eigenvalue estimates and geometric understanding.

Contribution

It introduces a refined Gershgorin disk bound for multiple eigenvalues of complex matrices, extending previous real matrix results and exploring applications to special matrix classes.

Findings

Multiple eigenvalues of complex matrices are contained in smaller Gershgorin disks.

The results yield improved eigenvalue bounds in the real case.

Applications to normal and almost symmetric matrices demonstrate the method's versatility.

Abstract

Extending an earlier result for real matrices we show that multiple eigenvalues of a complex matrix lie in a reduced Gershgorin disk. One consequence is a slightly better estimate in the real case. Another one is a geometric application. Further results of a similar type are given for normal and almost symmetric matrices.