v2009.01.01 - Convex Optimization

548 APPENDIX A. LINEAR ALGEBRA
By similar reasoning,
λ(I + µA) = 1 + λ(µA) (1355)
For vector σ(A) holding the singular values of any matrix A
σ(I + µA T A) = π(|1 + µσ(A T A)|) (1356)
σ(µI + A T A) = π(|µ1 + σ(A T A)|) (1357)
where π is the nonlinear permutation-operator sorting its vector
argument into nonincreasing order.
For A∈ S M and each and every ‖w‖= 1 [176,7.7, prob.9]
w T Aw ≤ µ ⇔ A ≼ µI ⇔ λ(A) ≼ µ1 (1358)
[176,2.5.4] (confer (37))
A is normal ⇔ ‖A‖ 2 F = λ(A) T λ(A) (1359)
For A∈ R m×n A T A ≽ 0, AA T ≽ 0 (1360)
because, for dimensionally compatible vector x , x T A T Ax = ‖Ax‖ 2 2 ,
x T AA T x = ‖A T x‖ 2 2 .
For A∈ R n×n and c∈R
tr(cA) = c tr(A) = c1 T λ(A)
(A.1.1 no.4)
For m a nonnegative integer, (1744)
det(A m ) =
tr(A m ) =
n∏
λ(A) m i (1361)
i=1
n∑
λ(A) m i (1362)
i=1