Linear Algebra, Theory And Applications, 2012a

324 SELF ADJOINT OPERATORS
and so from (13.18),
(
r∑
r∑
)
RUx = b k F x k = F b k x k
k=1
Is F ( ∑ r
k=1 b kx k )=F (x)?
( ( r∑
)
( r∑
) )
F b k x k − F (x) ,F b k x k − F (x)
k=1 k=1
k=1
=
=
=
=
( ( r∑
) ( r∑
))
(F ∗ F ) b k x k − x , b k x k − x
k=1
k=1
( ( r∑
) ( r∑
))
U 2 b k x k − x , b k x k − x
k=1 k=1
( ( r∑
) ( r∑
))
U b k x k − x ,U b k x k − x
k=1
k=1
)
r∑
b k Ux k − Ux, b k Ux k − Ux =0
( r∑
k=1
Because from (13.19), Ux = ∑ r
k=1 b kUx k . Therefore, RUx = F ( ∑ r
k=1 b kx k )=F (x).
The following corollary follows as a simple consequence of this theorem. It is called the
left polar decomposition.
k=1
Corollary 13.6.3 Let F ∈L(X, Y ) and suppose n ≥ m where X is a Hilbert space of
dimension n and Y is a Hilbert space of dimension m. Then there exists a Hermitian U ∈
L (X, X) , and an element of L (X, Y ) ,R,such that
F = UR, RR ∗ = I.
Proof: Recall that L ∗∗ = L and (ML) ∗ = L ∗ M ∗ . Now apply Theorem 13.6.2 to
F ∗ ∈L(Y,X). Thus,
F ∗ = R ∗ U
where R ∗ and U satisfy the conditions of that theorem. Then
F = UR
and RR ∗ = R ∗∗ R ∗ = I.
The following existence theorem for the polar decomposition of an element of L (X, X)
is a corollary.
Corollary 13.6.4 Let F ∈L(X, X). Then there exists a Hermitian W ∈L(X, X) , and
a unitary matrix Q such that F = WQ, and there exists a Hermitian U ∈L(X, X) and a
unitary R, such that F = RU.
This corollary has a fascinating relation to the question whether a given linear transformation
is normal. Recall that an n × n matrix A, is normal if AA ∗ = A ∗ A. Retain the same
definition for an element of L (X, X) .
Theorem 13.6.5 Let F ∈L(X, X) . Then F is normal if and only if in Corollary 13.6.4
RU = UR and QW = WQ.