LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.

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Generalized Ritz values obey min-max bounds and are monotone under trial-space inclusion

Statement

Let $A = A^{*} \in C^{n \times n}$ and $B = B^{*} > 0$ . Write the generalized eigenvalues of $(A, B)$ in increasing order $λ_{1} \leq \dots \leq λ_{n}$ . For a subspace $S$ with a full-column-rank $B$ -orthonormal basis matrix $Q_{S}$ , let $θ_{1} (S) \leq \dots \leq θ_{\dim S} (S)$ be the eigenvalues of $Q_{S}^{*} A Q_{S}$ . These values are independent of the chosen $B$ -orthonormal basis. Then, for $1 \leq j \leq \dim S$ , $λ_{j} \leq θ_{j} (S) .$ If $S \subseteq T$ and $j \leq \dim S$ , then $θ_{j} (T) \leq θ_{j} (S) .$

Facts & Assumptions

Given: A Hermitian matrix $A$ , a Hermitian positive-definite matrix $B$ , and full-column-rank $B$ -orthonormal basis matrices for the stated subspaces.

[F1]

The B-inner product, B-norm, and dual residual norm: $⟨ x, y ⟩_{B} = y^{*} B x$ and $∥ x ∥_{B}^{2} = x^{*} B x$ .

[F2]

A Hermitian-definite pencil has a real spectrum and a B-orthonormal eigenbasis: a Hermitian-definite pair $(A, B)$ has real generalized eigenvalues and a $B$ -orthonormal eigenbasis.

Proof

technique · direct

1.1F1F2algebra

For $Q_{S}$ , the compression $H_{S} = Q_{S}^{*} A Q_{S}$ is Hermitian. Diagonalize it with an orthonormal eigenbasis $z_{1}, \dots, z_{s}$ and ordered eigenvalues $θ_{1} (S) \leq \dots \leq θ_{s} (S)$ ; the vectors $Q_{S} z_{i}$ are B-orthonormal and every Rayleigh quotient on $S$ is their weighted average. Any j-dimensional $W \subseteq S$ intersects $span (Q_{S} z_{j}, \dots, Q_{S} z_{s})$ nontrivially by dimension count, so its maximum quotient is at least $θ_{j} (S)$ ; equality is attained on $span (Q_{S} z_{1}, \dots, Q_{S} z_{j})$ . This proves the min-max formula, and a unitary change between B-orthonormal bases conjugates the compression, proving basis invariance.

2.1F1F2step 1.1algebra

Let $u_{1}, \dots, u_{n}$ be the ambient B-orthonormal generalized eigenbasis from [F2], ordered by $λ_{i}$ . Any j-dimensional $W \subseteq S$ intersects $span (u_{j}, \dots, u_{n})$ nontrivially, again by dimension count. The quotient of a nonzero vector in that span is a weighted average of $λ_{j}, \dots, λ_{n}$ and is at least $λ_{j}$ ; the min-max formula from step 1.1 gives $θ_{j} (S) \geq λ_{j}$ .

3.1step 1.1algebra∎

If $S \subseteq T$ , every j-dimensional subspace considered for $θ_{j} (S)$ is also available for $θ_{j} (T)$ . The minimum over the larger family therefore cannot be greater, so $θ_{j} (T) \leq θ_{j} (S)$ .

Depends on

Used by

Dependency tree · two levels

13 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

Peter Arbenz, Lecture Notes on Solving Large Scale Eigenvalue Problems, §§13.1 and 13.6 (standard reference, not scraped)
Andrew Knyazev, Recent implementations, applications, and extensions of LOBPCG, §1 (standard reference, not scraped)