Theorems inverse SVD

Question 1

Let A ∈ C^m×n
If X is any matrix satisfying AXA = A then Ax = b has a
solution if and only if

Answer 1

AXb = b, in which case the general solution is
x = Xb + (I − XA)y

where y ∈ C^n is an arbitrary vector.

Question 2

Singular value decomposition

Answer 2

A matrix A ∈ C^m×n has a singular value decomposition:

A = UΣV*

where:
U ∈ C^m×m, V ∈ C^n×n -unitary 
Σ = diag(σ1, . . . , σp) ∈ R^m×n
p = min(m, n),
where σ1 ≥ σ2 ≥ · · · ≥ σp ≥ 0.
If A is real, U and V can be taken real orthogonal

Question 3

If A = UΣV* ∈ C^m×n SVD then:

Answer 3

A+ = VΣ+U*

where Σ+ = diag(σ−1_1…., σ−1_r, 0, . . . , 0) is n × m
and r = rank(A).

Question 4

Let the SVD of A ∈ C^m×n be given
U = [u1, . . . , um] and V = [v1, . . . , vn]

If k < r = rank(A)
Ak =Σ^k_i=1: σ_iu_iv*_i
then

Answer 4

min ||A-B||2 =||A-Ak||2 =σk+1.

rank(B)=k

Question 5

Minimum 2-norm solution

Answer 5

For given A ∈ C^m×n and b ∈ C^m with b ∈ range(A)

The vector x = A^+b is the solution of minimum 2-norm amongst all the solutionsto Ax = b

Question 6

Least squares solutions

Answer 6

For given A ∈ C^m×n and b ∈ C^m, the vectors
x = A+b + (I − A^+A)y y ∈ C^n arbitrary,

minimize ||Ax − b||2

Moreover xLS = A^+b is the least squares solution of minimum 2-norm