Chapter 15
Exponential Functions of Matrices
In Chap. 12, we dealt with a function of matrices. In this chapter we study several
important definitions and characteristics of functions of matrices. If elements of
matrices consist of analytic functions of a real variable, such matrices are of
particular importance. For instance, differentiation can naturally be defined with
the functions of matrices. Of these, exponential functions of matrices are widely used
in various fields of mathematical physics. These functions frequently appear in a
system of differential equations. In Chap. 10, we showed that SOLDEs with suitable
BCs can be solved using Green’s functions. In the present chapter, in parallel, we
show a solving method based on resolvent matrices. The exponential functions of
matrices have broad applications in group theory that we will study in Part IV. In
preparation for it, we study how the collection of matrices forms a linear vector
space. In accordance with Chap. 13, we introduce basic notions of inner product and
norm to the matrices.
15.1 Functions of Matrices
We consider a matrix in which individual components are differentiable with respect
to a real variable such that
A t
ð Þ ¼ a ij t
ð Þ
À
Á
:
ð15:1Þ
We define the differentiation as
A
0 t
ð Þ ¼
dA t
ð Þ
dt
a
0
ij t
ð Þ
:
ð15:2Þ
We have
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_15
581