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

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Hermitian eigenvalues lie within a certified Frobenius bound

Statement

If $n \geq 1$ , $A = A^{*} \in C^{n \times n}$ and $M \geq ∥ A ∥_{F}$ , then every eigenvalue $λ$ of $A$ satisfies $∣ λ ∣ \leq M$ . More generally, if a nonempty Hermitian matrix family $A \subseteq C^{n \times n}$ satisfies $∥ A ∥_{F} \leq M$ for all $A \in A$ , then every eigenvalue of every member lies in $[- M, M]$ .

Facts & Assumptions

Given: $A, A, M$ as in the statement.

[F1]

A Hermitian matrix has a unit eigenvector for each eigenvalue (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F2]

The operator 2-norm is at most the Frobenius norm (Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison).

[F3]

The operator 2-norm is induced by the Euclidean vector norm (The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||).

Proof

1.1givenF1F2F3algebra

Let $A v = λ v$ with $∥ v ∥_{2} = 1$ . Then $∣ λ ∣ = ∥ λ v ∥_{2} = ∥ A v ∥_{2} \leq ∥ A ∥_{2} \leq ∥ A ∥_{F} \leq M$ , using [F1], [F2], and the induced norm definition [F3].

2.1givenF1step 1.1∎

The same calculation applies separately to each $A \in A$ , using its unit eigenvectors and the common bound $M$ . Since Hermitian eigenvalues are real by [F1], all are in $[- M, M]$ .

Depends on

Used by

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Sources

Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §§5.1 and 5.6 (standard reference, not scraped)