Weyl Bounds
From the book
Chapter 7: Symmetric Matrix. In the chapter mind map this icon labels Weyl's Inequality: Eigenvalue Bounds of Sums. The discussion below is excerpted and lightly edited from § Theorem: Eigenvalue Bounds of Sums in Mathematics for AI and Machine Learning.
Let $A,B\in\mathbb{R}^{N\times N}$ be symmetric. Then every eigenvalue of $A+B$ lies in $[\lambda_{\min}(A+B), \lambda_{\max}(A+B)]$, where
__Proof:__ Take a unit eigenvector $\mathbf x$ of $A+B$ for $\lambda_{\min}(A+B)$; then $\lambda_{\min}(A+B) = \mathbf x^\top A\mathbf x + \mathbf x^\top B\mathbf x \ge \lambda_{\min}(A)+\lambda_{\min}(B)$ by @eq:rayleigh-quotient-bounds, and likewise for $\lambda_{\max}$.
What this drawing shows
What you see. Represents eigenvalue perturbation bounds that relate spectrum changes to matrix perturbations.
In the mind map. Chapter 7 — Weyl's Inequality: Eigenvalue Bounds of Sums. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 7: Symmetric Matrix — see the mind-map node Weyl's Inequality: Eigenvalue Bounds of Sums.
