(8) Let exp tA be a real orthogonal matrix with any real number t. Then, A is a real
skew-symmetric matrix.
(9) Let exp tA be a unitary matrix with any real number t. Then, A is an antiHermitian matrix.
To show (8), from the assumption with any real number t we have
exp tA exp tA
ð
Þ
T ¼ exp tA exp tA
T
À Á ¼ E:
ð15:51Þ
Differentiating (15.51) with respect to t by use of (15.4), we have
A exp tA
ð
Þexp tA
T
À Á þ exp tA
ð
ÞA
T exp tA
T
À Á ¼ 0:
ð15:52Þ
Since (15.52) must hold with any real number, (15.52) must hold with t ! 0 as
well. On this condition, from (15.52) we get A + A
T
¼ 0. That is, A is a real skewsymmetric matrix.
In the case of (9), similarly we have
exp tA exp tA
ð
Þ
{ ¼ exp tA exp tA
{
À Á ¼ E:
ð15:53Þ
Differentiating (15.53) with respect to t and taking t ! 0, we get A + A
{
¼ 0. That
is, A is an anti-Hermitian matrix.
Properties (6)–(9) including Properties (6)
0 and (7)
0 are frequently used later in
Chap. 20 in relation to Lie groups and Lie algebras.
15.3 System of Differential Equations
In this section we describe important applications of the exponential functions of
matrices to systems of differential equations.
15.3.1 Introduction
In Chap. 10 we have dealt with SOLDEs using Green’s functions. In the present
section we examine the properties of systems of differential equations. There is a
close relationship between the two topics. First we briefly outline it.
We describe an example of the system of differential equations. First, let us think
of the following general equation:
_
x t
ð Þ ¼ a t
ð Þx t
ð Þ þ b t
ð Þy t
ð Þ,
ð15:54Þ
592
15 Exponential Functions of Matrices