DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-16

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Triangularisable endomorphisms and simultaneous triangularisability

Definition

Let $V$ be a finite-dimensional vector space over $F$ . An endomorphism $T : V \to V$ is triangularisable over $F$ when some ordered basis $B$ makes $[T]_{B}^{B}$ upper triangular (Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases).

A family $A$ of endomorphisms of $V$ is simultaneously triangularisable over $F$ when one ordered basis $B$ makes $[S]_{B}^{B}$ upper triangular for every $S \in A$ . The empty family is simultaneously triangularisable in every ordered basis, including the empty basis of the zero space.

Depends on

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

Used by

Complete invariant flags are equivalent to upper-triangular matrices Proposition

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

S. Axler, Linear Algebra Done Right, 4th ed., Results 5.38-5.39 (standard reference, not scraped)