4 I. A Review of Linear Algebra
to
these bases is aij =
ft
Aej
=
(Ii,
Aej).
This
suggests
that
we
may
say
that
the
matrix
of
A relative
to
these
bases is
F*
AE.
In
this
notation,
composition
of
linear
operators
can
be
identified
with
matrix
multiplication as follows.
Let
M
be
a
third
Hilbert space
with
or-
thonormal
basis G =
(gl,""
gp).
Let
B E
.c(JC,
M).
Then
(matrix
of
B . A)
G*(B·
A)E
G*BF
F*AE
(G*
BF)(F*
AE)
(matrix
of
B)
(matrix
of
A).
The
second
step
in
the
above chain is justified by Exercise 1.1.2.
The
adjoint
of
an
operator
A E .c(H,
JC)
is
the
unique
operator
A*
in
.c(JC,
H)
that
satisfies
the
relation
(z,Ax)K: = (A*z,x)7t
for all x E
Hand
z E
JC.
Exercise
1.2.1
For fixed bases
in
Hand
JC,
the
matrix
of
A*
is the con-
jugate transpose
of
the
matrix
of
A.
For
the
space .c(H, H) we use
the
more compact
notation
.c(H).
In
the
rest
of
this
section,
and
elsewhere in
the
book, if no qualification is
made,
an
operator
would
mean
an
element
of
.c(H).
An
operator
A is called
self-adjoint
or
Hermitian
if A = A
*,
skew-
Hermitian
if A =
-A
*,
unitary
if
AA
* = I = A *
A,
and
normal
if
AA* = A*A.
A
Hermitian
operator
A is said
to
be
positive
or
positive
semidefinite
if (x,
Ax)
~
0 for all x E
H.
The
notation
A
~
0 will
be
used
to
express
the
fact
that
A is a positive
operator.
If
(x,
Ax)
> 0 for all nonzero x,
we
will say A is
positive
definite,
or
strictly
positive.
We will
then
write
A >
O.
A positive
operator
is
strictly
positive if
and
only if
it
is invertible.
If
A
and
B are
Hermitian,
then
we
say
A
~
B if A - B
~
O.
Given
any
operator
A we
can
find
an
orthonormal
basis Yl,
...
,Yn such
that
for each k =
1,2,
...
, n,
the
vector AYk is a linear
combination
of
Yl,
...
,Yk·
This
can
be
proved by
induction
on
the
dimension n
of
H.
Let
Al
be
any
eigenvalue
of
A
and
Yl
an
eigenvector corresponding
to
AI,
and
M
the
I-dimensional subspace
spanned
by it. Let N be
the
orthogonal com-
plement of
M.
Let
PN
denote
the
orthogonal projection
on
N.
For
YEN,
let
ANY
= PNAy.
Then,
AN
is a linear
operator
on
the
(n-I)-dimensional
space N. So, by
the
induction
hypothesis,
there
exists
an
orthogonal
ba-
sis
Y2,""
Yn
of
N such
that
for k = 2,
...
, n
the
vector ANYk is a linear
combination of
Y2,
...
,Yk. Now
Yl,
...
,Yn is
an
orthogonal basis for H,
and
each AYk
is
a linear combination
of
Yl,
...
,
Yk
for k =
1,2,
...
,n.
Thus,
the
matrix
of
A
with
respect
to
this
basis is
upper
triangular. In
other
words,
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