Chemistry Reference
In-Depth Information
Two matrices
A
and
B
of the same order are equal if
B ¼ A
B
ij
¼
A
ij
for all i
;
j
ð
2
3
Þ
:
Matrices can be added (or subtracted) if they have the same order:
A B ¼ C
C
ij
¼
A
ij
B
ij
ð
2
4
Þ
:
Addition and subtraction enjoy commutative and associative
properties.
Multiplying amatrix
A
by a complex number c impliesmultiplicationof
all elements of
A
by that number:
c
A ¼ B
B
ij
¼
cA
ij
ð
2
:
5
Þ
The product, rows by columns, of two (or more) matrices
A
by
B
is
possible if the matrices are conformable (the number of columns of
A
equals the number of rows of
B
):
C
ij
¼
X
n
AB ¼ C
m
nn
pm
p
A
i
a
B
a
j
;
ABC ¼ D
m
nn
pp
qm
q
;
a¼
1
ð
2
6
Þ
:
D
ij
¼
X
X
p
n
A
i
a
B
ab
C
b
j
a¼
1
b¼
1
Matrix multiplication is usually not commutative, the quantity
½A
;
B¼ABBA
ð
2
7
Þ
:
being the commutator of
A
and
B
.If
½A
;
B¼
0
ð
2
8
Þ
:
then matrices
A
and
B
commute.
The product of more than twomatrices enjoys the associative property:
ABC ¼ðABÞC ¼ AðBCÞ
ð
2
9
Þ
:
The trace of a product of matrices is invariant under the cyclic
permutation of its factors.
