18 Matrices and linear transformations
18.1 Concepts
Matrix algebra is the branch of mathematics concerned with the theory of systems of
linear equations. Just as ordinary algebra is the essential tool for the manipulation and
solution of single equations (or small numbers of equations), matrix algebra is used
for the manipulation of systemsof equations, and for the construction of numerical
methods of solution of the problems that give rise to such systems in the physical
sciences, in engineering, and in statistics. The economy and simplicity of the matrix
formalism also makes it the ideal tool for the theoretical analysis of the properties
and structure of systems of equations, and of the physical problems that lead to
them. One of the important applications in chemistry makes use of the concept of
linear transformation for the matrix representation of symmetry operations in the
description of the symmetry properties of molecules, of molecular wave functions,
and of normal modes of vibration. In this chapter, the elements of matrix algebra are
presented in Sections 18.2 to 18.4, and the matrix theory of linear transformations
in Sections 18.5 and 18.6. Symmetry operations are discussed in Section 18.7, with a
brief introduction to symmetry groups.
A matrix consists ofm 1 × 1 nquantities, or elements, arranged in a rectangular array
made up of mrows and ncolumns, and enclosed in parentheses.
1Examples are
(18.1)
and the general notation for an m 1 × 1 nmatrix (read as ‘m by n matrix’), with mn
elements, is (see Section 17.1 for determinants)
(18.2)
A=
aaa a
aaa a
aaa a
nnn11 12 13 121 22 23231 32 333aaa a
m
mm1 mn
2321
03
1234−
,
,
abc
def
x
x
x
x
,
100
010
001
1The term matrix was coined by Sylvester in 1850 to denote ‘an oblong arrangement of terms consisting,
suppose, of mlines and ncolumns’ because from it ‘we may form various systems of determinants’. James Joseph
Sylvester (1814–1897), studied mathematics at Cambridge but was ineligible for a Cambridge degree on religious
grounds (he was a Jew); he received his degree from Trinity College, Dublin. His appointments include the chair
of mathematics at the newly founded Johns Hopkins University in Baltimore (1876–1883). He was the founding
Editor of the American Journal of Mathematics (1878). He is best known for his collaboration with his friend
Cayley on the theory of matrices and forms.