DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passaudited 2026-08-28

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.

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The singular values of a linear map as the eigenvalues of the positive square root of T^*T

Definition

Let $T : V \to W$ be a linear map between finite-dimensional real or complex inner product spaces. Because

$⟨ T^{*} T v, v ⟩ = ⟨ T v, T v ⟩ \geq 0$

for every $v \in V$ , and because $(T^{*} T)^{*} = T^{*} T$ by Adjoints satisfy $(S + T)^{*} = S^{*} + T^{*}$ , $(λ T)^{*} = \overline{λ} T^{*}$ , $(S T)^{*} = T^{*} S^{*}$ , and $T^{* *} = T$ , the operator $T^{*} T$ is non-negative. Let

$∣ T ∣ : = \sqrt{T^{*} T},$

the unique non-negative square root from A non-negative operator has a unique non-negative square root. The singular values of $T$ are the eigenvalues of $∣ T ∣$ , listed with multiplicity in weakly decreasing order.

In particular, this definition applies to endomorphisms by taking $W = V$ .

Definition references

Adjoints satisfy $(S + T)^{*} = S^{*} + T^{*}$ , $(λ T)^{*} = \overline{λ} T^{*}$ , $(S T)^{*} = T^{*} S^{*}$ , and $T^{* *} = T$

The singular values of a linear map as the eigenvalues of the positive square root of T^*T

Definition

Definition references

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