Chemistry Reference
In-Depth Information
~
c
¼
ð
c
11
c
12
Þ ¼
ð
c
1
c
2
Þ
ð
1
:
9
Þ
where the tilde
means interchanging columns by rows or vice versa (the
transposed matrix).
The linear inhomogeneous system:
(
A
11
c
1
þ
A
12
c
2
¼
b
1
ð
:
Þ
1
10
A
21
c
1
þ
A
22
c
2
¼
b
2
can be easily rewritten in matrix form using matrix multiplication rule
(1.3) as:
Ac
¼
b
ð
1
:
11
Þ
where c and b are 2
1 column vectors.
Equation (1.10) is a system of two algebraic equations linear in the
unknowns c
1
and c
2
, the elements of matrix A being the coefficients of the
linear combination. Particular importance has the case where b is pro-
portional to c through a number
l
:
Ac
¼
l
c
ð
1
:
12
Þ
which is known as the eigenvalue equation for matrix A.
is called an
eigenvalue and c an eigenvector of the square matrix A. Equation (1.12) is
equally well written as the homogeneous system:
l
ð
A
l
1
Þ
c
¼
0
ð
1
:
13
Þ
where 1 is the 2
2 diagonal matrix having 1 along the diagonal, called
the identity matrix, and 0 is the zero vector matrix, a 2
1 column of
zeros. Written explicitly, the homogeneous system (Equation 1.13) is:
(
ð
A
11
l
Þ
c
1
þ
A
12
c
2
¼
0
ð
1
:
14
Þ
þð
l
Þ
¼
A
21
c
1
A
22
c
2
0
Elementary algebra then says that the system of equations (1.14) has
acceptable solutions if and only if the determinant of the coefficients
vanishes, namely if:
¼
A
11
l
A
12
j
A
l
1
j¼
0
ð
1
:
15
Þ
l
A
21
A
22
Equation (1.15) is known as the secular equation for matrix A.Ifwe
expand the determinant according to the rule of Equation (1.2), we obtain
