DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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The Gershgorin disks of an endomorphism with respect to an ordered basis

Definition

Let $V$ be a finite-dimensional complex vector space, let $T : V \to V$ be linear, and let $B = (b_{1}, \dots, b_{n})$ be an ordered basis. Write

$[T]_{B}^{B} = (a_{i j}) .$

For each $1 \leq i \leq n$ , the $i$ -th Gershgorin disk of $T$ with respect to $B$ is

$D_{i} (T, B) : = {z \in C : ∣ z - a_{i i} ∣ \leq \sum_{j \neq i} ∣ a_{i j} ∣} .$

The union $⋃_{i = 1}^{n} D_{i} (T, B)$ is the Gershgorin region of $T$ in the basis $B$ .

Depends on

Coordinate columns $[v]_{\mathcal B}$ and matrices $[T]_{\mathcal B}^{\mathcal C}$ of linear maps relative to ordered bases

Used by

Every eigenvalue lies in some Gershgorin disk Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sheldon Axler, Linear Algebra Done Right, fourth edition (standard reference, not scraped)