Next, we extend (15.46) to the exponential functions of matrices. Let us define the
differentiation of an exponential function of a matrix with respect to a real parameter
t. Then, in concert with (15.46), we have a following important theorem.
Theorem 15.4 [1, 2] Let F(t) exp tA with t being a real number and A being a
matrix that does not depend on t. Then, we have
d
dt
F t
ð Þ ¼ AF t
ð Þ:
ð15:47Þ
In (15.47) we assume that individual matrix components of F(t) are differentiable
with respect to t and that the differentiation of a matrix is defined as in (15.2).
Proof We have
F t þ Δt
ð
Þ¼ exp t þ Δt
ð
ÞA ¼ exp tA þ ΔtA
ð
Þ¼ exp tA
ð
Þ exp ΔtA
ð
Þ
¼ F t
ð ÞF Δt
ð Þ ¼ exp ΔtA
ð
Þ exp tA
ð
Þ¼F Δt
ð ÞF t
ð Þ,
ð15:48Þ
where we considered that tA and ΔtA are commutative and used Theorem 15.2.
Therefore, we get
1
Δt
F t þ Δt
ð
ÞÀF t
ð Þ
½
¼
1
Δt
F Δt
ð Þ À E
½
F t
ð Þ
¼
1
Δt
X 1
ν¼0
1
ν!
ΔtA
ð
Þ
ν À E
h
i
F t
ð Þ
¼
X 1
ν¼1
1
ν!
Δt
ð Þ
νÀ1 A
ν
h
i
F t
ð Þ:
ð15:49Þ
Taking the limit of both sides of (15.49), we get
lim
Δt!0
1
Δt
F t þ Δt
ð
ÞÀF t
ð Þ
½
d
dt
F t
ð Þ ¼ AF t
ð Þ:
Notice that in (15.49) only the first term (i.e., ν ¼ 1) of RHS is nonvanishing with
respect to Δt ! 0. Then, (15.47) will follow. This completes the proof.
∎
On the basis of the power series expansion of the exponential function of a matrix
defined in (15.7), A and F(t) are commutative. Therefore, instead of (15.47) we may
write
d
dt
F t
ð Þ ¼ F t
ð ÞA:
ð15:50Þ
Using Theorem 15.4, we show the following important properties. These are
converse propositions to Properties (6) and (7).
15.2 Exponential Functions of Matrices and Their Manipulations
591
differentiation of an exponential function of a matrix with respect to a real parameter
t. Then, in concert with (15.46), we have a following important theorem.
Theorem 15.4 [1, 2] Let F(t) exp tA with t being a real number and A being a
matrix that does not depend on t. Then, we have
d
dt
F t
ð Þ ¼ AF t
ð Þ:
ð15:47Þ
In (15.47) we assume that individual matrix components of F(t) are differentiable
with respect to t and that the differentiation of a matrix is defined as in (15.2).
Proof We have
F t þ Δt
ð
Þ¼ exp t þ Δt
ð
ÞA ¼ exp tA þ ΔtA
ð
Þ¼ exp tA
ð
Þ exp ΔtA
ð
Þ
¼ F t
ð ÞF Δt
ð Þ ¼ exp ΔtA
ð
Þ exp tA
ð
Þ¼F Δt
ð ÞF t
ð Þ,
ð15:48Þ
where we considered that tA and ΔtA are commutative and used Theorem 15.2.
Therefore, we get
1
Δt
F t þ Δt
ð
ÞÀF t
ð Þ
½
¼
1
Δt
F Δt
ð Þ À E
½
F t
ð Þ
¼
1
Δt
X 1
ν¼0
1
ν!
ΔtA
ð
Þ
ν À E
h
i
F t
ð Þ
¼
X 1
ν¼1
1
ν!
Δt
ð Þ
νÀ1 A
ν
h
i
F t
ð Þ:
ð15:49Þ
Taking the limit of both sides of (15.49), we get
lim
Δt!0
1
Δt
F t þ Δt
ð
ÞÀF t
ð Þ
½
d
dt
F t
ð Þ ¼ AF t
ð Þ:
Notice that in (15.49) only the first term (i.e., ν ¼ 1) of RHS is nonvanishing with
respect to Δt ! 0. Then, (15.47) will follow. This completes the proof.
∎
On the basis of the power series expansion of the exponential function of a matrix
defined in (15.7), A and F(t) are commutative. Therefore, instead of (15.47) we may
write
d
dt
F t
ð Þ ¼ F t
ð ÞA:
ð15:50Þ
Using Theorem 15.4, we show the following important properties. These are
converse propositions to Properties (6) and (7).
15.2 Exponential Functions of Matrices and Their Manipulations
591