8. Spectral Variation of Hermitian Matrices
Society for Industrial and Applied Mathematics eBooks · 2007
§28 Bounds in the Frobenius norm In 1934 Karl Löwner (later Charles Loewner) wrote a most remarkable paper that is widely known for initiating the theory of matrix monotone functions. In this paper Löwner states, without proof, the Frobenius norm analogue of Theorem 8.5. As we have seen, this was subsequently generalized in two different directions: Theorem 9.7 for Hermitian matrices asserting the same inequality for all unitarily-invariant norms, and Theorem 15.1 of Hoffman and Wielandt valid for normal matrices but restricted to the Frobenius norm. We present two proofs of Löwner's theorem that depend on ideas simpler than the ones needed for these more general versions. Lemma 28.1 Let x and y be any two vectors in . Then 〈 x↓ , y↑ 〉 ≤ 〈x,y〉 ≤ 〈 x↓ , y↓ 〉 . 28.1 Proof It is enough to prove this for . In this case the assertion is that whenever , and , then . The latter inequality can be written as and is obviously true. A matrix version of this is the following Proposition 28.2 Let A and B be n × n Hermitian matrices. Then 〈 Eig↓ (A), Eig↑ (B) 〉 ≤tr AB≤ 〈 Eig↓ (A), Eig↓ (B) 〉 . 28.2 Proof If A and B were commuting matrices, this would reduce to the preceding Lemma. The general case, in turn, can be reduced to this special one as follows.