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Topics in Matrix Analysis by Charles R. Johnson, Roger A. Horn

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5.5.12    Theorem. Let A, B ϵ Mn be given positive semidefinite Hermitian matrices. Arrange the eigenvalues of A B and B and the main diagonal entries di(A) of A in decreasing order λ1 ≥ · · · ≥ λn and d1(A) ≥ · · · ≥ dn(A). Then

image

    This follows directly from the special case of Theorem (5.5.19) in which A is positive semidefinite, since λi(B) = σi(B) when B is positive semidefinite. It also follows from Theorem (5.6.2) with image; see the discussion following the proof of Lemma (5.6.17).

Exercise. Deduce Theorem (5.5.11) from Theorem (5.5.12).

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