PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

Reduced QR gives the full-column and full-row-rank pseudoinverse formulas without forming normal equations

Statement

If $A \in M_{m \times n} (F)$ has full column rank and $A = \hat{Q} \hat{R}$ is a reduced QR factorisation, then $A^{+} = {\hat{R}}^{- 1} {\hat{Q}}^{*}$ .
If $A$ has full row rank and $A^{*} = \hat{Q} \hat{R}$ is a reduced QR factorisation of $A^{*}$ , then $A^{+} = \hat{Q} {\hat{R}}^{- *}$ .

Facts & Assumptions

Given: A compatible reduced QR factorisation.

[L1]

A reduced QR factorisation is $A = \hat{Q} \hat{R}$ with ${\hat{Q}}^{*} \hat{Q} = I$ and $\hat{R}$ square upper triangular (Full, reduced, and column-pivoted computational QR factorisations).

[L2]

In the full-column-rank case, $A^{+} = (A^{*} A)^{- 1} A^{*}$ (If $A$ has full column rank, then $A^{+} = (A^{*} A)^{- 1} A^{*}$ ).

[L3]

In the full-row-rank case, $A^{+} = A^{*} (A A^{*})^{- 1}$ (If $A$ has full row rank, then $A^{+} = A^{*} (A A^{*})^{- 1}$ ).

Proof

technique · direct

1.1

Suppose $A$ has full column rank and $A = \hat{Q} \hat{R}$ . By [L1], $A^{*} A = {\hat{R}}^{*} {\hat{Q}}^{*} \hat{Q} \hat{R} = {\hat{R}}^{*} \hat{R} .$ Using [L2], $A^{+} = (A^{*} A)^{- 1} A^{*} = ({\hat{R}}^{*} \hat{R})^{- 1} {\hat{R}}^{*} {\hat{Q}}^{*} = {\hat{R}}^{- 1} {\hat{Q}}^{*} .$

L1L2algebra

2.1

Suppose now that $A$ has full row rank and $A^{*} = \hat{Q} \hat{R}$ is a reduced QR factorisation of $A^{*}$ . Applying step 1.1 to $A^{*}$ gives $(A^{*})^{+} = {\hat{R}}^{- 1} {\hat{Q}}^{*} .$ Taking adjoints yields $A^{+} = ((A^{*})^{+})^{*} = \hat{Q} {\hat{R}}^{- *} .$

L1L3step 1.1algebra

3.1

These are exactly the reduced-QR pseudoinverse formulas.

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

12 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

Gene H. Golub, Least squares, singular values and matrix approximations (standard reference, not scraped)
Andrew Stuart and Jochen Voss, Matrix Analysis and Algorithms (standard reference, not scraped)