Chemistry Reference
In-Depth Information
Matrices
Matrices are the powerful algorithm connecting the differential equations
of quantum mechanics to equations governed by the linear algebra
of matrices and their transformations. After a short introduction on
elementary properties of matrices and determinants (Margenau and
Murphy, 1956; Aitken, 1958; Hohn, 1964), we introduce special matrices
and the matrix eigenvalue problem.
2.1 DEFINITIONS AND ELEMENTARY PROPERTIES
A matrix
A
of order m
n is an array of numbers or functions ordered
according to m rows and n columns:
0
@
1
A
A
11
A
12
A
1n
A
21
A
22
A
2n
A ¼
ð
2
1
Þ
:
A
m1
A
m2
A
mn
and can be denoted by its ij element (i
¼
1
2
;
...
;
m;
j
¼
1
2
;
...
;
n)as
;
;
A ¼f
A
ij
g
ð
2
2
Þ
:
Matrix
A
is rectangular if n
=
m and square if n
¼
m. In square
matrices, elements with j
¼
i are called diagonal. To any square matrix
A
we can associate two scalar quantities: its determinant
j
A
j¼
detA
(a number) and its trace tr
A ¼
P
i
A
ii
, the sum of all diagonal elements.
