Chemistry Reference
In-Depth Information
where A
ij
is a number called the ijth element of matrix A. The elements
A
ii
(j
i) are called diagonal elements. We are interested mostly in
symmetric matrices, for which A
21
¼
¼
A
12
.IfA
21
¼
A
12
¼
0, the matrix
is diagonal. Properties of a square matrix A are its trace
A
22
Þ;
the sum of its diagonal elements, and its determinant,denotedby
A
ð
tr A
¼
A
11
þ
jj¼
det A
;
a number that can be evaluated from its elements by the rule:
j
A
j¼
A
11
A
22
A
12
A
21
ð
1
:
2
Þ
Two 2
2 matrices can be multiplied rows by columns by the rule:
AB
¼
C
ð
1
:
3
Þ
!
B
11
B
12
B
21
B
22
!
!
A
11
A
12
A
21
A
22
C
11
C
12
C
21
C
22
¼
ð
1
:
4
Þ
the elements of the product matrix C being:
(
C
11
¼
A
11
B
11
þ
A
12
B
21
;
C
12
¼
A
11
B
12
þ
A
12
B
22
;
ð
1
:
5
Þ
C
21
¼
A
21
B
11
þ
A
22
B
21
;
C
22
¼
A
21
B
12
þ
A
22
B
22
:
So, we are led to the matrix multiplication rule:
X
2
C
ij
¼
A
i
k
B
k
j
ð
1
:
6
Þ
k¼
1
If matrix B is a simple number a, Equation (1.6) shows that all elements
of matrix A must be multiplied by this number. Instead, for a|A|, we have
from Equation (1.2):
¼
;
aA
11
aA
12
aA
11
A
12
aA
21
A
22
j
j¼
ð
Þ¼
ð
:
Þ
a
A
a
A
11
A
22
A
12
A
21
1
7
A
21
A
22
so that, multiplying a determinant by a number is equivalent to multi-
plying just one row (or one column) by that number.
We can have also rectangular matrices, where the number of rows is
different from the number of columns. Particularly important is the 2
1
column vector c:
!
¼
!
c
11
c
21
c
1
c
2
c
¼
ð
1
:
8
Þ
or the 1
2 row vector
~
c:
