Basics of matrices

Abstract

Theorem 19.1 (Weyl). Denote $\lambda _1 (C) \le \lambda _2 (C) \le \cdots \le \lambda _n (C)$ the (real) eigenvalues of an N ×N Hermitian matrix C. Let A,B be N ×N Hermitian matrices. Then, for any j ? {1,…, N},

$$\lambda _j (A) + \lambda _j (B) \le \lambda _j (A + B) \le \lambda _j (A) + \lambda _N (B).$$

In particular,

$$|\lambda _j (A + B) - \lambda _j (A)| \le ({\rm Tr}(B^2 ))^{\frac{1}{2}} .$$

((19.1))