Let us consider a matrix 
and a function of complex variable
There is a lot of possisilities to define a function of matrix f(A)
starting from the function of complex variable f(z). The
simplest of these possibilities seems to be the substituting the variable
"z" by the variable "A". For example,
and
as well
It turns out that this approach is not very practical for solving problems.
Definition 2.9.1. If
, f(z) is analytic in the open domain 
, 
is a closed simple line (does not cut itself) in
and the spectrum 
of the matrix A is included in domain 
enfolded by ,
then
where the integral is applied to the matrix by its elements.
Remark 2.9.1. Formula (28) is an analogue to the Cauchy integral formula proved for the functions of complex variable.
Example 2.9.1. Let f(z)=z
and 
Check how to calculate by rule (28). Since 
and 
are the eigenvalues of A ,
then let us choose the line 
where 
The function f(z)=z is analytic in domain 
First we find
and then
Proposition 2.9.1.
If 
then
where 
is a vector in space 
whose k-th component is one and the remaining ones are zeros.
Proof. Let 
Then,
Since the matrix is integrated by elements, we obtain that
Proposition 2.9.2. If
, there 
i.e., the conditions of definition 2.9.1
are satisfied, and
then
Proof. From (28) , (29) and XX-1=I
we find that
Since
and
then
that was to be proved.
Proposition 2.9.3. If
and 
is the Jordan normal form
of the matrix A , where
is an 
Jordan block, 
and f(z) is analytic on an open set that includes the spectre
of the matrix A, then
where
Prooof. Using Proposition 2.9.2
it is sufficient to consider only the value F(G), where 
is a 
Jordan block and 
Let the matrix zI-G be regular. Since
then
and
Now, taking into account the condition 
the assertion holds.
Example 2.9.2. Find 
if
Since 
and the function 
is analytic in the neighbourhood of 0 , then for the calculation of
we can apply the algorithm given in Proposition
2.9.3. We use for the calculation of Jordan
decomposition of the matrix A the ''Maple'':
Therefore, J=diag(J1,J2),where
Using (3), we find the matrices 
and 
:
and
After that, using (2), we find the matrix wanted:
Corollary 2.9.1. If
, 
and there
then
Proof. This is a special case of Proposition
2.9.3. All the Jordan blocks
are 
Example 2.9.3. If 
are the eigenvalues of the matrix
and 
are the corresponding linearly
indipendent eigenvectors , i.e.,
generate a basis in 
then 
and from analyticity of 
in the whole finite part of complex plane it follows that
where 
and
Next we consider the problem arising in the approximation of the function f(A) by the function g(A). This kind of problem arises, for example, if we replace f(A) with its Taylor polynomial of degree q.
Proposition 2.9.4.
Let ,
where
is an
Jordan block and 
If the functions f(z) and g(z) are analytic
on an open set containing the spectre
of the matrix A, then
Proof. Choosing 
we
have
Using Proposition 2.9.3 and inequality
we find that
and thus, the assertion holds.
Example 2.9.4. Let
We estimate the difference 
Since 
and the functions 
and g(z)=z are analytic in the neighbourhood of .1,then
we can apply the estimation (33) obtained in Proposition 2.9.4. First,
we use ''Maple'' for finding the Jordan
decomposition of the matrix A:
Hence, there is only one Jordan
block in the Jordan decomposition
of the matrix A, i.e.
Second, we use ''Maple'' for finding the condition number of X:
Since
and
then, by estimation (30), we have that
It is known that matrix X in the Jordan
decomposition of A is not uniquely determined. We try to choose
the matrix X so that the condition
number k2(X) should be minimal. Applying the
Filipov algorithm to find the Jordan
decomposition of A (see Proposition
2.5.2.1), we obtain that
where
It turns out that
is also a Jordan decomposition
of A, where
Therefore, the best estimation we can have by Proposition
2.9.4 is
Otherwise, in this example for calculating 
there is applicable the algorithm given in Proposition
2.9.3. Using the formula (32), we have that
By formula (31) we calculate the value of the function
in question:
Hence,
and
As a result of this example, we can assert that estimation
(33) proved in Proposition 2.9.5 is quite rough.
Proposition 2.9.5.
If the Maclaurin expansion of the function f(z)
is convergent in the circle containing the spectrum
of the matrix 
then
Prove this assertion with an additioal assumption
that the matrix A has a basis consisting of its eigenvectors. In
this case, by Corollary 2.9.1,
Proposition 2.9.6.
If the Maclaurin series of the function f(z)
is convergent in the circle containing the spectrum
of the matrix
then
Proof. Let us define the matrix E(s)
by
If 
,
then
is analytic, and therefore,
where 
By
comparing the powers of the variable s in (34)
and (35), we conclude that 
has the form
If 
then
Exercise 2.9.1. Prove that for an arbitrary matrix
there hold
and
Exercise 2.9.2. Apply Proposition
2.9.6 to the estimation of the errors in the approximate equalities
and
Proposition 2.9.7 (Sylvester
theorem). If all eigenvalues
of the matrix
are different, then
or
where 
is the determinant obtained from the Vandermonde
determinant
by replacing the k-th row vector
by the vector
Example 2.9.4. Calculate 
if 
First we find the eigenvalues of A:
Then we use formula (36):
Now applying (37), we have
We solve this problem once more using the formula 
where S is the matrix formed of the eigenvalues
of A. Find the eigenvalues of A:
and
and the matrix
Hence
