A t
ð Þ þ B t
ð Þ
½
Š
0 ¼ A
0 t
ð Þ þ B
0 t
ð Þ,
ð15:3Þ
A t
ð ÞB t
ð Þ
½
Š
0 ¼ A
0 t
ð ÞB t
ð Þ þ A t
ð ÞB
0 t
ð Þ:
ð15:4Þ
To get (15.4), putting A(t)B(t) C(t) we have
c ij t
ð Þ ¼
X
k
a ik t
ð Þb kj t
ð Þ,
ð15:5Þ
where C(t) ¼ (c ij (t)), B(t) ¼ (b ij (t)). Differentiating (15.5), we get
c
0
ij t
ð Þ ¼
X
k
a
0
ik t
ð Þb kj t
ð Þ þ
X
k
a ik t
ð Þb
0
kj t
ð Þ,
ð15:6Þ
namely, we have C
0 (t) ¼ A
0 (t)B(t) + A(t)B
0 (t), i.e., (15.4) holds.
Analytic functions are usually expanded as an infinite power series. As an
analogue, a function of a matrix is expected to be expanded as such. Among various
functions of matrices, exponential functions are particularly important and have a
wide field of applications. As in the case of a real or complex number x, the
exponential function of matrix A can be defined as
exp A E þ A þ
1
2!
A
2
þ Á Á Á þ
1
ν!
A
ν
þ Á Á Á,
ð15:7Þ
where A is a (n, n) square matrix and E is a (n, n) identity matrix. Note that E is used
instead of a number 1 that appears in the power series expansion of exp x described
as
exp x 1 þ x þ
1
2!
x
2
þ Á Á Á þ
1
ν!
x
ν
þ Á Á Á:
Here we define the convergence of a series of matrix as in the following
definition:
Definition 15.1 Let A 0 , A 1 , Á Á Á, A ν , Á Á Á [A ν (a ij, ν ) (1 i, j n)] be a sequence of
matrices. Let a ij, 0 , a ij, 1 , Á Á Á, a ij, ν , Á Á Á be the corresponding sequence of each matrix
component. Let us define a series of matrices as e
A e a ij
À Á ¼ A 0 þ A 1 þ Á Á Á þ
A ν þ Á Á Á and a series of individual matrix components as e a ij a ij,0 þ a ij,1 þ Á Á Á þ
a ij,ν þ Á Á Á. Then, if each e a ij converges, we say that e
A converges.
Regarding the matrix notation in Definition 15.1, readers are referred to (11.38).
Now, let us show that the power series of (15.7) converges [1, 2].
Let A be a (n, n) square matrix. Putting A ¼ (a ij ) and max
i, j
j a ij j¼ M and defining
A
ν
a
ν
ð Þ
ij
, where a
ν
ð Þ
ij is defined as a (i, j) component of A
ν , we get
582
15 Exponential Functions of Matrices