ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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The singular value decomposition of an explicit 2x3 matrix

Example

For the linear map $R^{3} \to R^{2}$ with matrix

$A = (\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 1 & 0 \end{array}),$

an SVD is

$A = U Σ V^{T}, U = I_{2}, Σ = (\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 1 & 0 \end{array}), V = I_{3} .$

Facts & Assumptions

Given: The matrix $A$ above with the standard inner products on $R^{3}$ and $R^{2}$ .

[L1]

Every linear map between finite-dimensional inner product spaces admits a singular value decomposition (Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition).

Verification

technique · direct

1.1

Direct multiplication gives $A^{T} A = diag (4, 1, 0)$ , so the singular values are $2, 1, 0$ , with standard basis vectors as right-singular vectors.

L1algebra

2.1

The images of the first two standard basis vectors are $2 e_{1}$ and $e_{2}$ , so the standard basis of $R^{2}$ gives the left-singular vectors. Therefore $U = I_{2}$ , $V = I_{3}$ , and the displayed diagonal matrix $Σ$ give an SVD of $A$ , exactly as [L1] guarantees.

L1step 1.1algebra∎

Depends on

Used by

The rank-one truncation of an SVD realises the Eckart-Young minimiser Example

Dependency tree · two levels

10 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)