DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-31

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The truncated singular-value pseudoinverse obtained by discarding singular values below a declared numerical-rank threshold

Definition

Let $F \in {R, C}$ , let $A \in M_{m \times n} (F)$ have singular value decomposition $A = U Σ V^{*}$ , and let $τ \geq 0$ be a declared threshold. The truncated singular-value pseudoinverse at threshold $τ$ is

$A_{τ}^{+} : = V Σ_{τ}^{+} U^{*},$

where $Σ_{τ}^{+} \in M_{n \times m} (F)$ is the transposed-shape diagonal matrix whose $i$ th diagonal entry is $σ_{i}^{- 1}$ when $σ_{i} > τ$ and is $0$ when $σ_{i} \leq τ$ ; every off-diagonal entry is $0$ .

The threshold is part of the definition, exactly as in Numerical rank relative to a declared norm, scale, and tolerance: changing $τ$ changes which singular directions are retained.

Depends on

Used by

Dependency tree · two levels

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Sources

Gene H. Golub, Least squares, singular values and matrix approximations (standard reference, not scraped)