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