TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix

Statement

Let $T : V \to W$ be a linear map between finite-dimensional inner product spaces. In orthonormal bases $E = (e_{j})$ of $V$ and $F = (f_{i})$ of $W$ ,

$[T^{*}]_{E \leftarrow F} = {\overline{[T]_{F \leftarrow E}}}^{T} .$

Over $R$ this is the transpose. The statement includes zero-sized bases.

Facts & Assumptions

Given: A map $T : V \to W$ and orthonormal bases $E = (e_{j})$ and $F = (f_{i})$ .

[L1]

The adjoint satisfies $⟨ T e_{j}, f_{i} ⟩ = ⟨ e_{j}, T^{*} f_{i} ⟩$ (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ , Every linear map between finite-dimensional inner product spaces has a unique adjoint).

[L2]

If $(e_{i})_{i < r}$ is an orthonormal basis, then $v = \sum_{i < r} ⟨ v, e_{i} ⟩ e_{i}$ for every vector $v$ (Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis).

[L3]

Matrix columns record the coordinates of images of basis vectors (Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases).

[L4]

Transposition interchanges matrix rows and columns, and complex conjugation is an involution (The transpose $A^{T}$ of a matrix, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

Proof

technique · direct

1.1L2L3

Write $A = [T]_{F \leftarrow E}$ and $B = [T^{*}]_{E \leftarrow F}$ . Applying [L2] in $F$ expands $T e_{j} = \sum_{i} ⟨ T e_{j}, f_{i} ⟩ f_{i}$ , and applying it in $E$ expands $T^{*} f_{i} = \sum_{j} ⟨ T^{*} f_{i}, e_{j} ⟩ e_{j}$ . Since a coordinate column in a basis is unique, [L3] gives $A_{i j} = ⟨ T e_{j}, f_{i} ⟩$ and $B_{j i} = ⟨ T^{*} f_{i}, e_{j} ⟩$ .

2.1step 1.1L1L4

By [L1] and conjugate symmetry, $B_{j i} = \overline{⟨ e_{j}, T^{*} f_{i} ⟩} = \overline{⟨ T e_{j}, f_{i} ⟩} = \overline{A_{i j}}$ . Thus [L4] gives $B = {\overline{A}}^{T}$ .

3.1step 2.1∎

If either basis is empty, the same entrywise identity is vacuous and identifies the unique matrix of the required size.

Depends on

Used by

Dependency tree · two levels

23 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, 4th ed., result 7.9 (standard reference, not scraped)
Sergei Treil, Linear Algebra Done Wrong, §5.5.1 (standard reference, not scraped)