Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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FALSE: The isometry in the polar decomposition is unique even for singular operators

Statement

The isometry in the polar decomposition is unique even for singular operators.

Facts & Assumptions

Given: The singular matrix T=(1000).

[L1]

This matrix has two distinct polar isometric factors S1 and S2 with the same non-negative factor U (A singular matrix has a polar decomposition with visibly nonunique isometric factors).

Refutation

technique · direct
1.1

By [L1], the singular matrix T admits both decompositions T=S1U and T=S2U, with S1≠S2.

L1
2.1

Therefore the isometry in a polar decomposition need not be unique for a singular operator. This refutes the claim.

L1∎

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