=+5.26. Show that A ≥ B implies |A|≥|B|. (Hint: Without loss of generality, let B > 0. Then A ≥ B iff B−1/2AB−1/2 ≥ I.)

5.27. Use the facts that for any symmetric matrix A, we have λmin(A) =

inf|x|=1(x

Ax/x

x) and λmax(A) = sup|x|=1(x

Ax/x

x) to prove the following string of inequalities. For any symmetric matrices A, B, we have

λmin(A) + λmin(B)

≤ λmin(A + B)

≤ λmin(A) + λmax(B)

≤ λmax(A + B)

≤ λmax(A) + λmax(B).