ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A diagonal rank-deficient matrix makes $A^{+}$ , $A A^{+}$ , and $A^{+} A$ completely explicit

Example

Let

$A = [\begin{array}{cc} 2 & 0 \\ 0 & 0 \end{array}] .$

Then

$A^{+} = [\begin{array}{cc} \frac{1}{2} & 0 \\ 0 & 0 \end{array}], A A^{+} = A^{+} A = [\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}] .$

So both $A A^{+}$ and $A^{+} A$ are the coordinate projection onto the first axis.

Facts & Assumptions

Given: The displayed diagonal matrix $A$ .

[L1]

Every finite real or complex matrix has a unique Moore--Penrose pseudoinverse (Every finite real or complex matrix has a unique Moore--Penrose pseudoinverse).

[L2]

$A A^{+}$ and $A^{+} A$ are the orthogonal projections onto $im A$ and $im A^{*}$ ( $A A^{+}$ and $A^{+} A$ are the orthogonal projections onto $im A$ and $im A^{*}$ ).

Verification

technique · computation

1.1L1algebra

Because $A$ is already diagonal with one nonzero singular value, inverting that entry and leaving the zero entry fixed gives the candidate $A^{+} = diag (1 / 2, 0)$ .

2.1step 1.1algebra

Direct multiplication gives $A A^{+} = A^{+} A = diag (1, 0)$ .

3.1L2step 2.1algebra∎

The matrix $diag (1, 0)$ is the orthogonal projection onto the first coordinate axis, which is exactly $im A = im A^{*}$ . This matches [L2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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A diagonal rank-deficient matrix makes A+A+, AA+AA+, and A+AA+A completely explicit