ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-08-29

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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All exterior powers of a diagonal operator are diagonal

Example

Let $T : R^{n} \to R^{n}$ be diagonal with diagonal entries $λ_{1}, \dots, λ_{n}$ . Then for every $0 \leq k \leq n$ , the map $Λ^{k} T$ is diagonal in the increasing wedge basis: for a $k$ -subset $I = {i_{1} < \dots < i_{k}}$ ,

$Λ^{k} T (e_{I}) = λ_{I} e_{I}, λ_{I} : = λ_{i_{1}} \dots λ_{i_{k}} .$

For instance, if $T = diag (2, 3, 5)$ on $R^{3}$ , then $Λ^{2} T$ is $diag (6, 10, 15)$ in the basis $(e_{1} \land e_{2}, e_{1} \land e_{3}, e_{2} \land e_{3})$ .

Facts & Assumptions

Given: A diagonal operator $T$ with diagonal entries $λ_{1}, \dots, λ_{n}$ and a $k$ -subset $I$ .

[L1]

Exterior powers preserve the induced maps on the wedge basis (Exterior powers are functorial).

[L2]

In the wedge bases, the matrix of $Λ^{k} T$ has entries $\det A_{J, I}$ , the signed $k$ -minors of the matrix $A$ of $T$ (In basis-wedge coordinates, the matrix of $Λ^{k} T$ is the signed matrix of $k$ -minors).

Verification

technique · direct

1.1given

The matrix $A$ of $T$ in the standard basis is diagonal.

2.1L2step 1.1algebra

By [L2], the entry of $Λ^{k} T$ at $(J, I)$ is the determinant of the $k \times k$ submatrix with rows $J$ and columns $I$ ; for the diagonal matrix of step 1.1 this is $0$ when $J \neq I$ and $λ_{I}$ when $J = I$ .

3.1L1step 2.1

By [L1], the values of $Λ^{k} T$ on the wedge basis $e_{I}$ determine the whole map, so $Λ^{k} T (e_{I}) = λ_{I} e_{I}$ and the map is diagonal with the displayed entries; the $n = 3$ case reads $diag (6, 10, 15)$ on $Λ^{2} R^{3}$ .

4.1step 2.1step 3.1∎

Steps 2.1 and 3.1 verify the diagonal form of every exterior power and the concrete example.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Keith Conrad, Exterior Powers (standard reference, not scraped)