Database Reference
In-Depth Information
Readers familiar with optimization theory will notice that the objective of (
8.8
),
although convex in each of the decision variables
X
and
Y
, is non-convex. There-
fore, a global solution by means of optimization algorithms is hard, if not impos-
sible. As foreshadowed in the introduction, however, (
8.8
) is equivalent to a well-
studied algebraic problem, namely, that of a
spectral decomposition
.
nn
be symmetric,
i.e.
,A¼ A
T
. Then there is a unique
Proposition 8.1
Let A
∈R
real sequence
λ
1
... λ
n
such that
U
T
,
A ¼ U
Λ
ð
8
:
9
Þ
Λ
ij
:
¼ δ
ij
λ
i
and U is unitary (
i.e.
,U
T
U ¼ UU
T
where
λ
i
are called
eigenvalues
or
spectrum
of A, the corresponding columns of U
eigenvectors,
and
the factorization
(
8.9
) eigenvalue
or
spectral decomposition
of A.
(Proofs as well as more detailed renditions of this result may be found in any
textbook on linear algebra. See, e.g., Chap. 1 of [HJ85].)
Given a matrix
A
¼ I). The values
mn
, the corresponding
Gram matrix
is defined as
∈R
:¼ A
T
A
G
:
Since
x
T
Gx ¼ x
T
A
T
Ax ¼
2
A
kk
0,
G
is positive semidefinite, which, as is well known in linear algebra, implies that its
spectrum is nonnegative. The same holds for the
covariance matrix
:¼ AA
T
C
:
Moreover, both the Gram as well as the covariance matrices are symmetric.
Now let a spectral decomposition of
G
be given by
V
T
,
G ¼ V
Λ
ð
8
:
10
Þ
and define
Z
:¼ AV
:
Then we obtain
Z
T
Z ¼ V
T
A
T
AV ¼ Λ:
Since
Λ
is a diagonal matrix of the eigenvalues of G
,
we may write
Z ¼ US
,i
:
e
:
AV ¼ US
ð
8
:
11
Þ
mm
and
S
mn
defined as
for some unitary
U
∈R
∈R
