As already shown in (15.7) of Sect. 15.1, an exponential function of a matrix, i.e.,
exp tA can be defined as
exp tA E þ tA þ
1
2!
tA
ð Þ
2 þ Á Á Á þ
1
ν!
tA
ð Þ
ν þ Á Á Á:
ð15:73Þ
From Property (5) of (15.41), we have det(exp A) 6 ¼ 0. This is because no matter
what number (either real or complex) TrA may take, exp t(TrA) never vanishes. That
is, F(t) ¼ exp A is non-singular, and so an inverse matrix of exp A must exist. This is
the case with exp tA as well; see equation below
5
ð Þ
0 det exp tA
ð
Þ¼ exp Tr tA
ð Þ ¼ exp t TrA
ð
Þ:
ð15:74Þ
Note that if A (or tA) is a (n, n) matrix, F(t) of (15.72) is a (n, n) matrix as well.
Since in that general case F(t) is non-singular, it consists of n linearly independent
column vectors, i.e., (n, 1) matrices; see Chap. 11. Then, F(t) is symbolically
described as
F t
ð Þ ¼ x 1 t
ð Þ x 2 t
ð ÞÁ Á Áx n t
ð Þ
ð
Þ ,
where x i (t) (1
i
n) represents a i-th column vector. Note that since F
(0) ¼ E, x i (0) has its i-th row component of 1, otherwise 0. Since F(t) is
non-singular, n column vectors x i (t) (1 i n) are linearly independent. In the
case of, e.g., (15.66), A is a (2, 2) matrix, which implies that x(t) and y(t) represent
two linearly independent column vectors, i.e., (2, 1) matrices. Furthermore, if we
look at the constitution of (15.70), x(t) and y(t) represent two linearly independent
solutions of (15.70). We will come back to this point later.
On the basis of the method of variation of constants, we deal with inhomogeneous
equations and describe below the procedures for solving the system of equations
with two unknowns. But, they can be readily conformed to the general case where
the equations with n unknowns are dealt with.
Now, we rewrite (15.68) symbolically such that
_
x t
ð Þ ¼ x t
ð ÞA,
ð15:75Þ
where x(t) (x(t) y(t)), _
x t
ð Þ ð _
x t
ð Þ _
y t
ð ÞÞ, and A
a c
b d
. Then, the inhomogeneous equation can be described as
_
x t
ð Þ ¼ x t
ð ÞA þ b t
ð Þ,
ð15:76Þ
where we define the inhomogeneous term as b (p q). This is equivalent to
596
15 Exponential Functions of Matrices
exp tA can be defined as
exp tA E þ tA þ
1
2!
tA
ð Þ
2 þ Á Á Á þ
1
ν!
tA
ð Þ
ν þ Á Á Á:
ð15:73Þ
From Property (5) of (15.41), we have det(exp A) 6 ¼ 0. This is because no matter
what number (either real or complex) TrA may take, exp t(TrA) never vanishes. That
is, F(t) ¼ exp A is non-singular, and so an inverse matrix of exp A must exist. This is
the case with exp tA as well; see equation below
5
ð Þ
0 det exp tA
ð
Þ¼ exp Tr tA
ð Þ ¼ exp t TrA
ð
Þ:
ð15:74Þ
Note that if A (or tA) is a (n, n) matrix, F(t) of (15.72) is a (n, n) matrix as well.
Since in that general case F(t) is non-singular, it consists of n linearly independent
column vectors, i.e., (n, 1) matrices; see Chap. 11. Then, F(t) is symbolically
described as
F t
ð Þ ¼ x 1 t
ð Þ x 2 t
ð ÞÁ Á Áx n t
ð Þ
ð
Þ ,
where x i (t) (1
i
n) represents a i-th column vector. Note that since F
(0) ¼ E, x i (0) has its i-th row component of 1, otherwise 0. Since F(t) is
non-singular, n column vectors x i (t) (1 i n) are linearly independent. In the
case of, e.g., (15.66), A is a (2, 2) matrix, which implies that x(t) and y(t) represent
two linearly independent column vectors, i.e., (2, 1) matrices. Furthermore, if we
look at the constitution of (15.70), x(t) and y(t) represent two linearly independent
solutions of (15.70). We will come back to this point later.
On the basis of the method of variation of constants, we deal with inhomogeneous
equations and describe below the procedures for solving the system of equations
with two unknowns. But, they can be readily conformed to the general case where
the equations with n unknowns are dealt with.
Now, we rewrite (15.68) symbolically such that
_
x t
ð Þ ¼ x t
ð ÞA,
ð15:75Þ
where x(t) (x(t) y(t)), _
x t
ð Þ ð _
x t
ð Þ _
y t
ð ÞÞ, and A
a c
b d
. Then, the inhomogeneous equation can be described as
_
x t
ð Þ ¼ x t
ð ÞA þ b t
ð Þ,
ð15:76Þ
where we define the inhomogeneous term as b (p q). This is equivalent to
596
15 Exponential Functions of Matrices