ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

An exact rank-two CUR reconstruction from a nonsingular cross

Example

Take $A = [\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \end{array}]$ , and select $I = J = (1, 2)$ . The resulting skeleton CUR factors reconstruct $A$ exactly.

Facts & Assumptions

Given: The displayed matrix $A$ and ordered row and column selections $I = J = (1, 2)$ .

[A1]

The matrix and index lists are exactly those stated in the Example.

[F1]

Skeleton and projected least-squares CUR cores: $C = A (:, J)$ , $R = A (I, :)$ , $W = A (I, J)$ , and the skeleton reconstruction is $C W^{†} R$ .

[F2]

Skeleton CUR is exact for a full-rank cross: if $rank (A) = rank (W) = k$ , then $C W^{†} R = A$ .

[F3]

The Moore--Penrose pseudoinverse $A^{+}$ as the solution of the four Penrose equations: a Moore-Penrose inverse satisfies the four Penrose equations.

[F4]

Every finite real or complex matrix has a unique Moore--Penrose pseudoinverse: every finite matrix has a unique Moore-Penrose inverse.

Given: $A$ and $I, J$ from [A1].

Verification

1.1givenA1F1F3F4algebra

Direct extraction gives $C = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{array}]$ , $R = [\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \end{array}]$ , and $W = I_{2}$ . Since $I_{2}$ satisfies the four Penrose equations [F3], uniqueness [F4] gives $W^{†} = I_{2}$ ; [F1] therefore gives the reconstruction $C R$ .

2.1A1F2F3F4step 1.1algebra∎

Multiplication yields $C R = [\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \end{array}] = A$ . The identity $I_{2}$ satisfies the Penrose equations [F3], hence equals $W^{†}$ by uniqueness [F4]. The third row of $A$ is the sum of the first two, and the first two are independent, so $rank (A) = 2 = rank (W)$ . Hence [F2] also applies and confirms the exact skeleton CUR identity.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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