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Basics of matrices

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Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 1957)

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))

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Correspondence to Alice Guionnet .

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© 2009 Springer-Verlag Berlin Heidelberg

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Guionnet, A. (2009). Basics of matrices. In: Large Random Matrices: Lectures on Macroscopic Asymptotics. Lecture Notes in Mathematics(), vol 1957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69897-5_20

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