**5.5.12 Theorem.** Let *A, B* ϵ *M*_{n} be given positive semidefinite Hermitian matrices. Arrange the eigenvalues of *A* ∘ *B* and *B* and the main diagonal entries *d*_{i}(*A*) of *A* in decreasing order *λ*_{1} ≥ · · · ≥ *λ*_{n} and *d*_{1}(*A*) ≥ · · · ≥ *d*_{n}(*A*). Then

This follows directly from the special case of Theorem (5.5.19) in which *A* is positive semidefinite, since *λ*_{i}(*B*) = *σ*_{i}(*B*) when *B* is positive semidefinite. It also follows from Theorem (5.6.2) with ; 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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