v2009.01.01 - Convex Optimization

A.6. SINGULAR VALUE DECOMPOSITION, SVD 565
Square matrix Σ is diagonal (A.1.1)
δ 2 (Σ) = Σ ∈ R η×η (1452)
holding the singular values {σ i ∈ R} of A which are always arranged in
nonincreasing order by convention and are related to eigenvalues by A.14
⎧
⎨
σ(A) i = σ(A T ) i =
⎩
√
λ(AT A) i = √ (√
λ(AA T ) i = λ AT A)
i
)
= λ(√
AA
T
> 0, 1 ≤ i ≤ ρ
i
0, ρ < i ≤ η
(1453)
of which the last η −ρ are 0 , A.15 where
ρ ∆ = rankA = rank Σ (1454)
A point sometimes lost: Any real matrix may be decomposed in terms of
its real singular values σ(A) ∈ R η and real matrices U and Q as in (1450),
where [134,2.5.3]
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 (1455)
)
R{q i |σ i =0} ⊆ N(A)
A.6.2
Subcompact SVD
Some authors allow only nonzero singular values. In that case the compact
decomposition can be made smaller; it can be redimensioned in terms of rank
ρ because, for any A∈ R m×n
ρ = rankA = rank Σ = max {i∈{1... η} | σ i ≠ 0} ≤ η (1456)
There are η singular values. For any flavor SVD, rank is equivalent to
the number of nonzero singular values on the main diagonal of Σ.
A.14 When A is normal,
√
σ(A) = |λ(A)|. [344,8.1]
)
A.15 For η = n , σ(A) = λ(A T A) = λ(√
AT A where λ denotes eigenvalues.
√
)
For η = m , σ(A) = λ(AA T ) = λ(√
AA
T
.