# Gershgorin Disks for Multiple Eigenvalues of Non-negative Matrices

• Imre Bárány
• József Solymosi
Chapter

## Abstract

Gershgorin’s famous circle theorem states that all eigenvalues of a square matrix lie in disks (called Gershgorin disks) around the diagonal elements. Here we show that if the matrix entries are non-negative and an eigenvalue has geometric multiplicity at least two, then this eigenvalue lies in a smaller disk. The proof uses geometric rearrangement inequalities on sums of higher dimensional real vectors which is another new result of this paper.

## Notes

### Acknowledgements

This research was supported by ERC Advanced Research Grant no 267165 (DISCONV). Imre Bárány is partially supported by Hungarian National Research Grant K 111827. József Solymosi is partially supported by Hungarian National Research Grant NK 104183 and by an NSERC Discovery Grant. We are indebted to three anonymous referees for very useful comments and information that have improved the presentation of this paper.

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