Download 9.3. Infinite Series Of Matrices. Norms Of Matrices

9.3. Infinite Series Of Matrices.
Norms Of Matrices
Let A   aij  be an n  n matrix of scalar entries.
Given a scalar function f  x  , we
can construct a matrix- valued function f  A by replacing x with A in f  x  .
Hopefully, useful characteristics of f  x  would be carried over to f  A . For
example, we expect the exponential e A to possess the laws of exponents, namely,
t s A
for all scalars s, t.
(9.8)
etAesA  e 
O
e I
(9.9)
where O and I are the n  n zero and identity matrices, respectively. However, for
transcendental functions, the substitution scheme does not tell us how to calculate the
numerical value of f  A for a given A. Here, as in the scalar case, the problem is
solved by defining the function in terms of a (convergent) power series.

1 k
x
k 0 k !
ex  

For example,

1 k
A
k 0 k !
eA  
The next question naturally concerns the convergence of power series of matrices.
Definition:
Convergent Series Of Matrices
Given an infinite sequence of m  n matrices
 C  c  , the series of matrices
k 
ij
k

C
k 1
is convergent iff all mn series
k

 c 
k 1
for i  1,
k
ij
are convergent.
, m and j  1,
(9.10)
In which case, we write


A   Ck   aij 
aij   cij
with
k
k 1
k 1
Definition:
,n
Norm Of A Matrix
m n
Let M   be the space of all m  n matrices of scalar entries. The norm is a mapping
M
mn 
F
by
A
A
where F is the scalar field R or C, such that the following axioms are satisfied
(a)
A  0 with
A  0 iff A  O
[Positivity]
(b)
cA  c A
 cF
[Homogeneity]
(c)
A B  A  B
[Triangle inequality]
Comment
As in the case of vectors, there are many realization of the norm.
Let A   aij  be an m  n matrix of scalar entries.
m
We shall define
n
A   aij
(9.11)
i 1 j 1
Theorem 9.1.
Definition (9.11) is indeed a norm.
Furthermore,
AB  A B .
Proof
We shall only prove the inequality
AB  A B , which can be viewed as a generalized
form of the Cauchy- Schwarz inequality. Assuming A   aij  is m  n and B   bij 
is n  p, we have
p
m
AB  
i 1 j 1
p
m
m
n
a
k 1
n
n
ik bkj
p
n
  aik bkj
i 1 j 1 k 1
  aik blj
m
p
n
n
  aik blj  kl
i 1 j 1 k 1 l 1
 A B
i 1 j 1 k 1 l 1
Comment
In the special case where A  B are n  n square matrices, the C-S inequality becomes
A2  A
2
which is easily generalized to give
Ak  A
k
for all k  1,2,3,
Theorem 9.2.
Let
Test For Convergence Of Matrix Series
 C  c   be an infinite sequence of n  n matrices such that the series of norms
k 
ij
k

C
k 1

k
converges. Then the matrix series
C
k 1
k
also converges.
Proof
Obviously,
cij k  
n
 c  
l ,m 1
k
lm
Ck

Therefore, the convergence of the series

k 1

Ck
implies the convergence of
k 1

for all i and j. This in turn implies the convergence of
 c 
k 1
Definition (9.10) then proves the theorem.
 c 
k
ij
for all i and j.
k
ij