v2009.01.01 - Convex Optimization

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U T = U † for orthonormal (including the orthogonal) matrices U . So, for
orthonormal matrices U,Q and arbitrary A
(UAQ T ) † = QA † U T (1751)
E.0.0.0.2
Prove:
Exercise. Kronecker inverse.
(A ⊗ B) † = A † ⊗ B † (1752)
E.0.1
Logical deductions
When A is invertible, A † = A −1 ; so A † A = AA † = I . Otherwise, for
A∈ R m×n [127,5.3.3.1] [208,7] [256]
g. A † A = I , A † = (A T A) −1 A T , rankA = n
h. AA † = I , A † = A T (AA T ) −1 , rankA = m
i. A † Aω = ω , ω ∈ R(A T )
j. AA † υ = υ , υ ∈ R(A)
k. A † A = AA † , A normal
l. A k† = A †k , A normal, k an integer
Equivalent to the corresponding Moore-Penrose condition:
1. A T = A T AA † or A T = A † AA T
2. A †T = A †T A † A or A †T = AA † A †T
When A is symmetric, A † is symmetric and (A.6)
A ≽ 0 ⇔ A † ≽ 0 (1753)
E.0.1.0.1 Example. Solution to classical linear equation Ax = b .
In2.5.1.1, the solution set to matrix equation Ax = b was represented
as an intersection of hyperplanes. Regardless of rank of A or its shape
(fat or skinny), interpretation as a hyperplane intersection describing a
possibly empty affine set generally holds. If matrix A is rank deficient or
fat, there is an infinity of solutions x when b∈R(A). A unique solution
occurs when the hyperplanes intersect at a single point.