Chemistry Reference
In-Depth Information
Equation (1.44) is easily verified:
0
@
1
A
0
@
1
A
8
<
1
2
1
2
1
2
1
2
l
1
P
1
þ
l
2
P
2
¼ðaþbÞ
þðabÞ
1
2
1
2
1
2
1
2
ð
1
:
45
Þ
0
1
:
a
þ
b
2
a
b
2
a
þ
b
2
a
b
2
þ
!
¼
@
A
ab
ba
¼
¼
A
a
þ
b
2
a
b
2
a
þ
b
2
a
b
2
þ
The same holds true for any analytical function
4
F of matrix A:
F
ð
A
Þ¼
F
ð
l
Þ
P
1
þ
F
ð
l
Þ
P
2
ð
1
:
46
Þ
1
2
Therefore, it is easy to calculate, say, the inverse or the square root of
matrix A. For instance, we obtain for the inverse matrix
¼
1
ð
F
Þ
:
<
:
0
@
1
A
0
@
1
A
1
1
1
1
ðaþbÞ
ðaþbÞ
ðabÞ
ðabÞ
2
2
2
2
l
1
1
P
1
þ
l
2
1
P
2
¼
þ
1
1
1
1
2
ðaþbÞ
2
ðaþbÞ
2
ðabÞ
2
ðabÞ
!
!
ðabÞþðaþbÞðabÞðaþbÞ
ðabÞðaþbÞðabÞþðaþbÞ
2
a
2
b
1
1
¼
¼
b
2
b
2
2
ða
2
Þ
2
ða
2
Þ
2
b
2
a
!
a b
ba
1
A
1
¼
¼
b
2
a
2
ð
1
:
47
Þ
and we obtain the usual result for the inverse matrix
ð
A
1
A
¼
AA
1
¼
1
Þ:
l
p
and
l
p
are positive, we can calculate the
In the same way, provided
:
p
square root of matrix A F
¼
p
8
<
:
p
aþb
p
ab
¼
P
1
þ
P
2
0
@
1
A
A
þ
B
A
B
!
p
aþb
p
ab
p
aþb
p
2
2
ð
1
:
48
Þ
þ
ab
1
2
¼
¼
p
aþb
p
p
aþb
p
ab
A
B
A
þ
B
ab
þ
2
2
4
Any function expressible as a power series, e.g. inverse, square root, exponential.
