Linear Algebra, Theory And Applications, 2012a

328 SELF ADJOINT OPERATORS
where σ is of the form
⎛
⎞
σ 1 0
⎜
σ = ⎝
. ..
⎟
⎠
0 σ k
for the σ i the singular values of A, arranged in order of decreasing size.
Proof: By the above lemma and Theorem 13.3.3 there exists an orthonormal basis,
{v i } n i=1 such that A∗ Av i = σ 2 i v i where σ 2 i > 0fori =1, ··· ,k,(σ i > 0) , and equals zero if
i>k.Thus for i>k,Av i = 0 because
For i =1, ··· ,k, define u i ∈ F m by
Thus Av i = σ i u i . Now
(Av i ,Av i )=(A ∗ Av i , v i )=(0, v i )=0.
(u i , u j ) = ( σ −1
i
= ( σ −1
i
u i ≡ σ −1
i Av i .
Av i ,σ −1
j
) (
Av j = σ
−1
i v i ,σ −1
j A ∗ )
Av j
v i ,σ −1
j σ 2 ) σ j
jv j = (v i , v j )=δ ij .
σ i
Thus {u i } k i=1 is an orthonormal set of vectors in Fm . Also,
AA ∗ u i = AA ∗ σ −1
i
Av i = σ −1
i AA ∗ Av i = σ −1
i Aσ 2 i v i = σ 2 i u i .
Now extend {u i } k i=1 to an orthonormal basis for all of Fm , {u i } m i=1
and let
U ≡ ( )
u 1 ··· u m
while
V ≡ ( )
v 1 ··· v n .
Thus U is the matrix which has the u i as columns and V is defined as the matrix which has
the v i as columns. Then
⎛
u ∗ ⎞
1
.
U ∗ AV =
u ∗ k
A ( )
v 1 ··· v n
⎜ ⎟
⎝
. ⎠
u ∗ m
⎛
=
⎜
⎝
u ∗ 1
.
.
u ∗ k
.
.
u ∗ m
⎞
( σ1 u 1 ··· σ k u k 0 ··· 0 ) (
σ 0
=
⎟
0 0
⎠
where σ is given in the statement of the theorem.
The singular value decomposition has as an immediate corollary the following interesting
result.
)