v2009.01.01 - Convex Optimization

A.7. ZEROS 569
Given symmetric matrix A∈ S n and its diagonalization A = SΛS T
(A.5.2), its pseudoinverse simply inverts all nonzero eigenvalues;
A † = SΛ † S T (1464)
A.6.5
SVD of symmetric matrices
A.6.5.0.1 Definition. Step function. (confer4.3.2.0.1)
Define the signum-like quasilinear function ψ : R n → R n that takes value 1
corresponding to a 0-valued entry in its argument:
[
ψ(a) =
∆
lim
x i →a i
x i
|x i | = { 1, ai ≥ 0
−1, a i < 0 , i=1... n ]
∈ R n (1465)
Eigenvalue signs of a symmetric matrix having diagonalization
A = SΛS T (1448) can be absorbed either into real U or real Q from the full
SVD; [303, p.34] (conferC.4.2.1)
A = SΛS T = Sδ(ψ(δ(Λ))) |Λ|S T ∆ = U ΣQ T ∈ S n (1466)
△
or
A = SΛS T = S|Λ|δ(ψ(δ(Λ)))S T ∆ = UΣ Q T ∈ S n (1467)
where Σ = |Λ| , denoting entrywise absolute value of diagonal matrix Λ .
A.7 Zeros
A.7.1
norm zero
For any given norm, by definition,
‖x‖ l
= 0 ⇔ x = 0 (1468)
Consequently, a generally nonconvex constraint in x like ‖Ax − b‖ = κ
becomes convex when κ = 0.