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

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for any Hermitian A, B ϵ Mn and any α ϵ [0, 1].

23.    Let A(t) ϵ Mn be differentiable and nonsingular. Use (6.5.9) to prove that

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Use this to show that if A(t) is positive definite and dA(t)/dt is positive semidefinite, then det A(t) is a strictly increasing function of t.

24.    Let X, Y ϵ Mn be given.

  (a)    Use the Cauchy-Schwarz inequality for the Frobenius norm to show that

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  (b)    Let σ1(X) ≥ · · · ≥ σn(X) ≥ 0 denote the ordered singular ...

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