v2009.01.01 - Convex Optimization

566 APPENDIX A. LINEAR ALGEBRA
Now
⎡
q T1
A = UΣQ T = [u 1 · · · u ρ ] Σ⎣
.
⎤
∑
⎦ = ρ σ i u i qi
T
qρ
T i=1
(1457)
U ∈ R m×ρ , Σ ∈ R ρ×ρ , Q ∈ R n×ρ
where the main diagonal of diagonal matrix Σ has no 0 entries, and
R{u i } = R(A)
R{q i } = R(A T )
(1458)
A.6.3
Full SVD
Another common and useful expression of the SVD makes U and Q
square; making the decomposition larger than compact SVD. Completing
the nullspace bases in U and Q from (1455) provides what is called the
full singular value decomposition of A ∈ R m×n [287, App.A]. Orthonormal
matrices U and Q become orthogonal matrices (B.5):
R{u i |σ i ≠0} = R(A)
R{u i |σ i =0} = N(A T )
R{q i |σ i ≠0} = R(A T )
R{q i |σ i =0} = N(A)
For any matrix A having rank ρ (= rank Σ)
q T1
A = UΣQ T = [ u 1 · · · u m ] Σ⎣
.
⎡
q T n
⎤
∑
⎦ = η σ i u i qi
T
⎡
σ 1
= [ m×ρ basis R(A) m×m−ρ basis N(A T ) ] σ 2
⎢
⎣
...
i=1
(1459)
⎤
⎡ ( n×ρ basis R(A T ) ) ⎤
T
⎥⎣
⎦
⎦
(n×n−ρ basis N(A)) T
U ∈ R m×m , Σ ∈ R m×n , Q ∈ R n×n (1460)
where upper limit of summation η is defined in (1451). Matrix Σ is no
longer necessarily square, now padded with respect to (1452) by m−η