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13
MEM633 Lectures 4&5 2-3 Linear Operators Def
:^ functionLet^ X
and
Y^
be arbitrary nonempty sets and let
D
be a nonempty subset of
X
.^ A function
f^
from
D
into
Y
is a rule that to each
x^
in^
D^
assigns a
unique element
f(x)
in
Y.
Def
: domain, rangeThe domain of
f^
is the set
D^
on which the function
f^ is defined, often written as
D
(f). The range of
f
is the set
R (f)
f(x)
:^ x
in
D^
Def
: linear operator X ,^ Y
are linear spaces over
F. A function
T^
is
said to be a linear operator if and only if
T^
( c^1
(^1) x + c
(^2) x 2
c^1
T^ x
1 + c
^ T^ 2
(^2) x
for any vectors
(^1) x ,^ x
2
in^
X^
and any scalars
c^1
, c
2
in^ F
Example
Consider
the
operator
that
rotates
a
vector
in
a
geometric plane counterclockwise 90° with respectto the origin.
y
x (^1) x (^2) x
(^3) x
This operator is a linear operator. (Why?) Matrix Representation of aLinear Operator
X^ and
Y^
are
n - and
m
-dimensional vector spaces
over the same field
F. Let {
(^1) x, x
, x
n^ } be a basis
for
X^
and {
(^1) y, y
, y
m } a basis for
Y. Suppose the
linear operator T^ :
X^
→
Y
is defined by
Tx
i^ = a
y 1i 1 + a
y 2i
+ a
mi^
m y
i=1,2,...,n
where
a
1i^
,^ a
2i^
,a
mi^
are
scalars
in
F
15
Then the operator
T^
can be represented by a matrix
A^ :
T^ :
x
a^
y = Ax
with
A
= [
aij
] ,
i=1,2,...,m; j=1,2,...,n.
Example: Consider
the
rotation
operator
in
the
previous
example again. If we choose {
(^1) x, x
2 } as the basis of
both
X
and
Y
, then the matrix representation of
T
is
0
1 1
−^0 ⎡^
⎤ ⎢^
⎥ ⎣^
⎦
If we choose {
(^1) x , x
2 } as the basis of
X
and the
basis {
(^2) x , x
3 } of
Y , then the matrix representation of
T^
is
1
0 0
1 ⎡^
⎤ ⎢^
⎥ ⎣^
⎦
2-4 Linear Algebraic Equations
Consider the set of linear equations
a^11
x^1
+ a
x 12
+ a
x1n = yn^
1
a^21
x^1
+ a
x 22
+ ... + a 2
x2n
= yn^
2
:^
Am
x^1
+ a
m
x^2
+ ... + a
mn
x^ n^
= y
m
where the given
aij
's and
yi
's are assumed to be
elements of a field
F , the unknown
x^ j
' s
are also
required to be in the same field.
This set of equations can be written in matrix form as
Ax = y
where
A^
is an
mxn
matrix [
aij^
],^ x
is an
nx
vector
and
y^
is an
mx
vector. We can consider the matrix
A^ as a linear operator which maps
n F into
m F
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Def:
null space The null space of a linear operator
A
is the set
N (
A )^
defined by^ N
( A
)^ = {
x
in
n F :^ Ax=
i.e.,
N
( A
)^ is the set of all solutions of
Ax=
Example:
Consider the matrix
A
⎡^
⎢^
=^ ⎢
⎢^
⎣^
which maps
(^5) into
(^3)
. Let
x^
= [
x^1
x^2
... x
´ ] 5
, then
Ax
= x
a 1
1 + x
(^2) a _2
(^3) a _3
a 4
4 + x
(^5) a 5
= x
a 1
1 + x
a 2
2 + x
( a 3
1 + a
+ x
( a 4
(^2) a ) + x
( a 5
(^2) a )
= (x
+x 1
+x^3
+x 4
)^5
(^1) a + (x
+x 2
+2x 3
+3x 4
)^ a 5
2
The
vectors
a
1 and
a
2 are
linearly
independent,
hence
Ax = 0
if and only if x+x 1
+x^3
+x 4
x+x^2
+2x^3
+3x 4
It is clear that the set of the three vectors
-1 -1^100
-1 -2^010
,^
and
-1 -3^001
form a basis of
N
( A )
Theorem:^ Let
A^
be an
mxn
matrix, then
dim
R (
A )^
N
( A
)^ =
n.
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2-5 Eigenvalues and EigenvectorsDef
: eigenvalue and eigenvectorLet^ A
be a linear operator that maps
n into itself
and
x^
a vector in
n
. Then
x^
is an eigenvector of
A^ corresponding to the eigenvalue
λ^
if^ x
≠^0
and
Ax
=^ λ
x. To find an eigenvalue of
A , we write
Ax=
λ x^
as
(^ A
-^ λΙ
)^ x
=^
This equation has a nontrivial solution if and only ifdet
( A -
= 0. det
( A -
is a polynomial of degree
n
in^ λ
and is called the characteristic polynomial of
A.
Example:
A^
⎡^
=^ ⎢
The eigenvalues of
A
are
λ^1
= 2 and
λ^2
= 4 and
their corresponding eigenvectors are
(^1) x
= [ 1
1 ]´
and
x
2 = [1 -1 ]´.
Example:
Suppose
that
an
nxn
matrix
A
has
n
linearly
independent eigenvectors
(^1) x , x
, x
n^
correspond-
ing to eigenvalues
λ ,^1
λ^2
λ n^
. Then any vector
y
can be written in the form
y^ = a
(^1) x _1
(^2) x 2
+ a
x n n
where [
a^1
a^2
an^
´ ]is called the representation of
y
with respect to the basis {
(^1) x ,^ x
,^ x
n^ }. From the
fact that
A x
i =
λ i^
i x
it follows that^ A
k y^
λ^1
k^ ) a
(^1) x 1
λ^2
k^ ) a
(^2) x 2
λ n^
k^ ) a
n x n
If^
|^ λ^1
|^ > |
λ i^
|,^ i
2,3,...,n
and
a^1
then
k A y
will tend to lie along the vector
(^1) x when
k^
is large.
25
Def:
operator norm The norm of a linear operator
A^
is defined as
0
1
:^ sup
sup
x^
x Ax
A^
Ax
x ≠^
=
=^
where "sup" stands for supremum, the least upperbound.The operator norm
A^
is defined through the
vector norm
x
.^
Therefore, for different
x^ , we
have different
A
Theorem:
Let
A^
be an
mxn
matrix. Then
1
1 max(
|^
m
ij
j^
i
A^
a
=^
∑
and
1 max(
|^
n
ij
i^
j
A^
a
∞^
=
=^
∑
Theorem:^ Let
A^
be in
mxn
and the maximum eigenvalue of
* A
A^
is
. Then A^2
Def:
inner product The inner product of
x^
and
y^
in^
X^
is a function of
x^
and
y , denoted by
< x
, y^
> , which satisfies the
following axioms:(1)
,^
x^ y
y x
<^
>^.
,^
,^
x^
y z
x z
y z
<^
+^
>^.
,^
cx y
c^
x y
<^
>^ for all scalars
c.
,^
x^ x <
and
,^
x^ x <
if^
x^ ≠
.
Example:
In^
n , the inner product is defined as
1
n
i^ i i
x y
x y
x y =
<^
=^ ∑
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Theorem:
(Cauchy-Schwarz Inequality)
1/ 2^
1/ 2
,^
,^
x y
x x
y y
<^
>^
<^
The equality holds if and only if
x^
and
y
are
linearly dependent. Theorem:
, x x <^
>^
has the property of a norm.
Normed Linear Spaces and Inner-Product Spaces
Def
:^
normed linear spaces A linear space on which a norm is defined is calleda normed linear space. Example
The space
n equipped with the norm
||^ x
||^ p
= ( |x
p | (^1)
p | n 1/p)
≥p
is a normed linear space. The space is denoted by^ ( ) p^
n l^
Example
:^
l^ spaces p
Let
≤^
p <
. The space
l^ p
consists of all infinite
sequences of scalars { x
, x
..... , n
} such that
|^1
p | i i
x ∞ =
∑
The norm in
l^ is defined by p
1/ 1 :^
|^
p pi
p^
i x^
x ∞ ⎧ =
=^ ⎨
⎩^
∑
The space
l^ ∞
is defined to consist of the bounded
sequences { x
, x^ n
}, with the norm
:^ sup |
| i i x^
x =∞
31
Def
:^ orthogonal complementTwo
vectors
x
,^
y^
in^
a^
Hilbert
space
X
are
orthogonal if
x
,^ y
If^
M^
is a closed
subspace
of
X
,^
then
⊥ M
,^
the
orthogonal
complement of
M
, is the set of all vectors in
X
which are orthogonal to every vector in
M
. That is,
⊥ =^
X^
M^
M
Time-domain and Frequency-domainSpaces
Time-domain spaces
L^2
Consider a signal vector
x (t) = [ x
(t) x 1
(t) 2
x^ (t) ]n
where x
(t), i = 1,2,i^
...n, is a complex-valued function
defined for all time, -
∞^
< t <
. Assume that the
signals under consideration satisfy
2 1
n
i i
x t
dt
∞
= −∞
∑∫
The
space
of
all
such
signals
is
denoted
by
( L^2
n)(-
∞), or simply by
L^2
). This space is
a Hilbert space with inner-product
,^
( )^
x^ y
x t^
y t dt ∞ −∞
<^
∫
Then the norm of x is
1/ 2 2
2
1
n
i i
x^
x t
dt
=^ ⎜
⎝^
∑∫
Frequency-domain space
L^2
Consider a function
[^
]
1
2
(^
)^
(^
)^
(^
)^
...^
(^
T )
n
x j
x^
j^
x^
j^
x^
j
ω^
Where
(^
x^ j^ ω i
,^
i^ =
...n,
is
a
complex-valued
function defined for all frequencies, -
∞^
<^
ω^
<^
33
Assume
that
the
functions
under
consideration
satisfy
2
1
(^
n
i i
x^
j^
d
ω^
∞ =−∞
The
space
of
all
such
functions
is
denoted
by
( L^2
n ) , or simply by
L^2
. This space is a Hilbert space
with inner-product
1
,^
(^
)^
(^
x^ y
x j
y j
d
ω^
ω^
ω
∞ π−∞
<^
The norm on
L
is 2
1/ 2 2
2
1 1
(^
n
i i
x^
x^
j^
d
ω^
= −∞ ⎛^
=^ ⎜
⎝^
RL
space
the set of rational functions in
L^2
with real coeffi-
cients.
These
functions
are
strictly
proper
and
have no poles on the imaginary axis. ( H )^2
n , H
space 2
a^
closed
subspace
of
L
with
functions
x(s)
analytic in Re(s) > 0.
H^2
⊥^
space
the orthogonal complement of
H
in^
L^2
( L ∞
nxm)
, L
∞^
space
the space of nxm complex-valued matrix functionswhose
largest
singular
values
are
essentially
bounded. The
L
-norm of a matrix function∞
Φ(s)
in^
L ∞
is
[^
]
:^
sup
(^
ess
j
∞ Φ^
=^
RL
∞^
space
Φ(s)
RL
∞^
iff^
Φ(s) is proper rational with real
coefficients and has no poles on the imaginaryaxis. ( H ∞
nxm)
, H
∞^
space
a^
closed
subspace
of
L
∞^
with
functions
(s)
analytic in Re(s) > 0. RH ∞
space
the set of proper stable rational matrices with realcoefficients.
