Linear and multilinear algebra; matrix theory Eigenvalues and eigenvectors Theorem

Gershgorin circle theorem

Statement

Let $A = (a_{ij})$ be an $n \times n$ matrix over $\C$ . For $1 \le i \le n$ , let $R_i = \sum_{j \ne i} \abs{a_{ij}}$ , and let $D_i = \{z \in \C : \abs{z - a_{ii}} \le R_i\}$ be the closed disc with centre $a_{ii}$ and radius $R_i$ . Then each eigenvalue of $A$ lies in at least one of the discs $D_1, \dots, D_n$ .

Remarks

The discs are called Gershgorin discs. The proof takes an eigenvector $x$ and an index $i$ with $\abs{x_i}$ maximal, and estimates row $i$ of $Ax = \lambda x$ . Since $A$ and $A^T$ have the same eigenvalues, the same holds with column sums. A strictly diagonally dominant matrix, with $\abs{a_{ii}} > R_i$ for all $i$ , has no eigenvalue $0$ , so it is invertible.

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Vector space definition
Linear map definition
Endomorphism definition
Eigenvalue definition
Gershgorin circle theorem statement

Tag	`0ETGP0XVCJ`
URI	`m2://0ETGP0XVCJ`
Kind	statement, theorem
Writing	W1 stated
MSC	15A42, 15A18
Status	This copy of the site has no build data. Run `m2 status 0ETGP0XVCJ`.
Uses	Eigenvalue
Same as	Wikidata Q978688 Wikipedia Gershgorin circle theorem

Gershgorin circle theorem

Statement

Remarks

Reading path

Further reading

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