ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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An explicit Hermitian 2x2 matrix is unitarily diagonalised

Example

For

$H = (\begin{array}{cc} 2 & i \\ - i & 2 \end{array}),$

the unitary matrix

$U = \frac{1}{\sqrt{2}} (\begin{array}{cc} i & - i \\ 1 & 1 \end{array})$

satisfies

$U^{*} H U = diag (3, 1) .$

Facts & Assumptions

Given: The Hermitian matrix $H$ above, acting on $C^{2}$ with the standard Hermitian inner product.

[L1]

A normal operator on a finite-dimensional complex inner product space has an orthonormal eigenbasis (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

Verification

technique · direct

1.1

One checks that $H (i, 1)^{T} = 3 (i, 1)^{T}$ and $H (- i, 1)^{T} = (- i, 1)^{T}$ . After dividing by $\sqrt{2}$ , these eigenvectors are orthonormal, so the columns of $U$ form the orthonormal eigenbasis guaranteed by [L1].

L1algebra

2.1

Because the columns of $U$ are orthonormal eigenvectors with eigenvalues $3$ and $1$ , the conjugated matrix $U^{*} H U$ is $diag (3, 1)$ .

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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)