Weyl's inequality and eigenvalue perturbation

ProofAdvanced⏱ 30 minHigh-Dim Statisticsweylperturbationeigenvalues

Theory: Covariance Estimation and Sample Complexity →

Problem

Prove Weyl's inequality: for symmetric $A,E$ ,

\big|\lambda_k(A+E) - \lambda_k(A)\big| \le \|E\| \quad\text{for every } k .

Deduce that an operator-norm bound $\|\hat\Sigma-\Sigma\|\le\eps$ controls every eigenvalue of $\hat\Sigma$ simultaneously, and explain why the corresponding statement for eigenvectors is false.

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints

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Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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