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

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A sphere net controls the operator norm of a symmetric matrix

Statement

Let $A \in R^{p \times p}$ be symmetric and let $N$ be a $1 / 4$ -net of the Euclidean unit sphere in $R^{p}$ . Then $∥ A ∥_{o p} \leq 2 \max_{v \in N} ∣ v^{T} A v ∣ .$ The operator norm is the finite-dimensional norm of The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||.

Facts & Assumptions

Given: A symmetric real matrix $A$ and a $1 / 4$ -net $N \subseteq S^{p - 1}$ , with $p \geq 1$ .

[F1]

A symmetric matrix has an orthonormal eigenbasis (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis), and its operator norm equals its largest absolute eigenvalue (A symmetric matrix operator norm is its largest absolute Rayleigh quotient).

[F2]

Every point of the sphere is within $1 / 4$ in Euclidean norm of some point of $N$ (Finite epsilon nets of the Euclidean sphere).

[F3]

The Euclidean inner product satisfies Cauchy–Schwarz (The Cauchy-Schwarz inequality for finite sums).

[F4]

The operator norm bounds $∥ A w ∥_{2} \leq ∥ A ∥_{o p} ∥ w ∥_{2}$ (The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||).

Proof

1.1F1F2choose

By [F1], choose a unit eigenvector $u$ for an eigenvalue $λ$ with $∣ λ ∣ = ∥ A ∥_{o p}$ . By [F2], choose $v \in N$ with $∥ u - v ∥_{2} \leq 1 / 4$ .

2.1F1F3F4step 1.1algebra∎

Symmetry and [F3]–[F4] imply $∣ u^{T} A u - v^{T} A v ∣ \leq ∣ (u - v)^{T} A u ∣ + ∣ v^{T} A (u - v) ∣ \leq 2 (1 / 4) ∥ A ∥_{o p} .$ Thus $∣ v^{T} A v ∣ \geq ∣ λ ∣ - \frac{1}{2} ∥ A ∥_{o p} = \frac{1}{2} ∥ A ∥_{o p}$ , giving the claimed inequality.

Depends on

Used by

Dense sample covariance concentration for sub-Gaussian vectors Theorem

Dependency tree · two levels

38 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

Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Chapter 1, Lemma 1.18 (standard reference, not scraped)
John Hunter, Measure Theory, finite epsilon nets and Euclidean norm facts (standard reference, not scraped)