PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-23 (gpt-6.1-sol)

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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.

Whitened observation identifiability and stability

Statement

Let $E$ be a nonzero linear subspace of the finite coefficient space $R^{m}$ , let $G : R^{m} \to R^{p}$ be the declared linear observation map, and let $R \in R^{p \times p}$ be symmetric positive definite. Let $R^{1 / 2}$ be the unique symmetric positive-semidefinite square root of $R$ and set $W = (R^{1 / 2})^{- 1}$ , so $W^{T} W = R^{- 1}$ . The whitened map is $A = W (G ∣_{E}) : E \to R^{p}$ , and define its smallest gain by $σ_{\min} (A) = \inf {∥ A v ∥_{2} : v \in E, ∥ v ∥_{2} = 1}$ ; the singular-value decomposition shows this infimum is the smallest singular value and is attained. Identifiability on $E$ means injectivity of $G ∣_{E}$ . It holds exactly when $σ_{\min} (A) > 0$ . If it holds, then for every true $a \in E$ , every realized error $e \in R^{p}$ , and datum $y = G a + e$ , the unique weighted least-squares estimate $\hat{a} = \arg \min_{b \in E} ∥ W (y - G b) ∥_{2}$ satisfies $∥ \hat{a} - a ∥_{2} \leq \frac{∥ W e ∥_{2}}{σ_{\min} (A)} .$ If $σ_{\min} (A) = 0$ , a nonzero vector in $E \cap \ker G$ makes distinct coefficients observationally indistinguishable. The estimate is pointwise in the realized error; it is not a confidence or probability statement. For a pressure-field application, let $d \geq 1$ be an integer and let $Ω \subseteq R^{d}$ be a nonempty spatial region, let $R^{Ω}$ be the set of real-valued functions on $Ω$ , and let $Π : E \to R^{Ω}$ be the chosen linear pressure representation. An additive-constant coefficient direction is a nonzero $v_{c} \in E$ for which $Π (v_{c}) (x) = c$ for all $x \in Ω$ and some nonzero $c \in R$ . If also $G v_{c} = 0$ , then $G (a + t v_{c}) = G a$ for every $a \in E$ and $t \in R$ . Restricting the admissible coefficients to a linear complement of $span {v_{c}}$ fixes this direction; such a complement exists because $E$ is finite-dimensional. Alternatively, identify $a$ and $b$ exactly when $a - b \in span {v_{c}}$ and treat each such equivalence class as one coefficient. This finite-dimensional result is not a general coefficient-identifiability theorem for a partial differential equation.

Statement references

Finite coefficient observation and innovation interface

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

A non-negative operator has a unique non-negative square root

Axler, Linear Algebra Done Right, §7E, pp. 270–278

Law, Stuart, and Zygalakis, Chapter 2 §2.1 and Chapter 4 §4.1

Facts & Assumptions

Given: The finite coefficient observation map, a nonzero coefficient subspace $E$ , a symmetric positive-definite weighting matrix $R$ , and a realized vector $e$ with $y = G a + e$ .

[A1]

The coefficient and one-window observation maps have the finite-dimensional domains and matrix meanings stated in the local observation-interface definition. Finite coefficient observation and innovation interface

[F1]

Every finite-dimensional positive-semidefinite operator has a unique positive-semidefinite square root. A non-negative operator has a unique non-negative square root

[F2]

Every linear map between finite-dimensional real inner-product spaces has orthonormal domain and codomain bases in which it acts by its nonnegative singular values and is zero on the remaining domain basis vectors. Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition

Proof

Given: The hypotheses in the Statement.

1.1A1F1givenalgebra

Since $R ≻ 0$ , it is positive semidefinite, so [F1] gives a symmetric positive-semidefinite $S$ with $S^{2} = R$ . If $S v = 0$ , then $R v = S^{2} v = 0$ , and positive definiteness of $R$ forces $v = 0$ ; hence $S$ is invertible. Therefore $W = S^{- 1}$ is symmetric positive definite and $W^{T} W = S^{- 2} = R^{- 1}$ .

1.2F2givenalgebra

Apply [F2] to $A$ on the nonzero finite-dimensional inner-product space $E$ : for an orthonormal basis $e_{1}, \dots, e_{d}$ of $E$ , the SVD gives $r \leq \min (d, p)$ positive singular values $s_{1}, \dots, s_{r}$ , an orthonormal list $f_{1}, \dots, f_{r}$ in $R^{p}$ , $A e_{i} = s_{i} f_{i}$ for $i \leq r$ , and $A e_{i} = 0$ for $i > r$ . For $v = \sum_{i} c_{i} e_{i}$ with $∥ v ∥_{2} = 1$ , $∥ A v ∥_{2}^{2} = \sum_{i = 1}^{r} s_{i}^{2} c_{i}^{2}$ . Thus the infimum in the Statement is zero when $r < d$ and is $\min_{i} s_{i}$ when $r = d$ ; it is positive exactly when $A$ is injective. If $r < d$ , any $e_{j}$ with $j > r$ is a nonzero kernel vector.

2.1A1step 1.1step 1.2algebra

The operator $W$ is invertible by step 1.1, so for every $v \in E$ , $A v = 0$ if and only if $W G v = 0$ , which holds if and only if $G v = 0$ . Thus $A$ is injective exactly when $G ∣_{E}$ is injective, and in the noninjective case a nonzero $v \in E \cap \ker G$ gives $G (a + v) = G a$ for every $a \in E$ .

3.1A1F2step 1.2algebra∎

Suppose $σ_{\min} (A) > 0$ . Step 1.2 then has $r = d \leq p$ , so the $d$ vectors $f_{1}, \dots, f_{d}$ form an orthonormal list that extends to an orthonormal basis $f_{1}, \dots, f_{p}$ of $R^{p}$ by [F2]. For $y = G a + e$ , write $a = \sum_{i = 1}^{d} a_{i} e_{i}$ and $W y = \sum_{i = 1}^{d} b_{i} f_{i} + b_{⊥}$ , where $b_{⊥}$ is orthogonal to each $f_{i}$ for $i \leq d$ . For any $b = \sum_{i = 1}^{d} c_{i} e_{i} \in E$ , $∥ W (y - G b) ∥_{2}^{2} = \sum_{i = 1}^{d} (b_{i} - s_{i} c_{i})^{2} + ∥ b_{⊥} ∥_{2}^{2}$ ; since each $s_{i} > 0$ , this is uniquely minimized by $c_{i} = b_{i} / s_{i}$ . Therefore $\hat{a} - a = \sum_{i = 1}^{d} s_{i}^{- 1} ⟨ W e, f_{i} ⟩ e_{i}$ . Orthonormality gives $∥ \hat{a} - a ∥_{2}^{2} = \sum_{i = 1}^{d} s_{i}^{- 2} ∣ ⟨ W e, f_{i} ⟩ ∣^{2} \leq σ_{\min} (A)^{- 2} \sum_{i = 1}^{d} ∣ ⟨ W e, f_{i} ⟩ ∣^{2} \leq σ_{\min} (A)^{- 2} ∥ W e ∥_{2}^{2}$ , where the last inequality follows by expanding $W e$ in the full orthonormal basis. Taking square roots proves the bound.

Whitened observation identifiability and stability

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources