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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 positive diagonal Gram matrix gives a nontrivial polar factor

Example

For $A = (\begin{array}{cc} 0 & - 2 \\ 1 & 0 \end{array}),$ the Gram matrix is positive diagonal, its positive square root is nontrivial, and the polar isometry is a quarter-turn.

Facts & Assumptions

Given: The displayed real matrix $A$ .

[F1]

A nonnegative matrix has a unique nonnegative square root (A non-negative operator has a unique non-negative square root).

[F2]

The polar decomposition has positive factor $\sqrt{A^{*} A}$ and isometry factor $S$ with $A = S \sqrt{A^{*} A}$ (Every endomorphism has a polar decomposition T = SU with U non-negative and S an isometry on the orthogonal complement of ker T, and S is unique exactly when T is invertible).

Verification

1.1givenF1algebra

Direct multiplication gives $A^{*} A = (\begin{array}{cc} 1 & 0 \\ 0 & 4 \end{array})$ , and $H = (\begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array})$ is positive with $H^{2} = A^{*} A$ . By [F1], $H$ is the unique nonnegative square root.

2.1F2step 1.1algebra∎

Put $S = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ . Then $S^{*} S = I$ and $S H = A$ , so [F2] identifies $S$ as the polar isometry and $H$ as its positive factor.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)
Nicholas J. Higham, Computing the Polar Decomposition—with Applications (standard reference, not scraped)