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operation, is a commutative ring, as the reader can show. A third commutative ring is the
integers modulo
p
together with the operations of addition and multiplication modulo
p
.(See
Problem
6.2
.) The ring of matrices introduced in the next section is not commutative. Some
important commutative rings are introduced in Section
6.7.1
.
6.2.2 Matrices
A
matrix over a set
R
is a rectangular array of elements drawn from
R
consisting of some
number
m
of rows and some number
n
of columns. Rows are indexed by integers from the set
{
1, 2, 3,
...
,
m
}
{
1, 2, 3,
...
,
n
}
.Theentry
in the
i
th row and
j
th column of
A
is denoted
a
i
,
j
, as suggested in the following example:
and columns are indexed by integers from the set
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
a
1,1
a
1,2
a
1,3
a
1,4
1234
5678
9101112
A
=[
a
i
,
j
]=
a
2,1
a
2,2
a
2,3
a
2,4
=
a
3,1
a
3,2
a
3,3
a
3,4
Thus,
a
2,3
=
7and
a
3,1
=
9.
The
transpose
of a matrix
A
, denoted
A
T
, is the matrix obtained from
A
by exchanging
rows and columns, as shown below for the matrix
A
above:
⎡
⎤
159
2610
3711
4812
⎣
⎦
A
T
=
Clearly, the transpose of the transpose of a matrix
A
,
(
A
T
)
T
, is the matrix
A
.
A
column
n
-vector
x
is a matrix containing one column and
n
rows, for example:
⎡
⎤
⎡
⎤
x
1
x
2
.
x
n
5
6
.
8
⎣
⎦
⎣
⎦
x
=
=
A
row
m
-vector
y
is a matrix containing one row and
m
columns, for example:
y
=[
y
1
,
y
2
,
...
,
y
m
]=[
1, 5,
...
,9
]
The transpose of a row vector is a column vector and vice versa.
A
square matrix
is an
n
×
n
matrix for some integer
n
.The
main diagonal
of an
n
×
n
square matrix
A
is the set of elements
{
a
1,1
,
a
2,2
,
...
,
a
n−
1,
n−
1
,
a
n
,
n
}
. The diagonal below
(above) the main diagonal is the elements
{
a
2,1
,
a
3,2
,
...
,
a
n
,
n−
1
}
(
{
a
1,2
,
a
2,3
,
...
,
a
n−
1,
n
}
).
The
n
×
n
identity matrix
,
I
n
,isasquare
n
×
n
matrix with value 1 on the main diagonal
and 0 elsewhere. The
n
n
zero matrix
,0
n
, has value 0 in each position. A matrix is
upper
(
lower
)
triangular
if all elements below (above) the main diagonal are 0. A square matrix
A
is
symmetric
if
A
=
A
T
,thatis,
a
i
,
j
=
a
j
,
i
for all 1
×
≤
i
,
j
≤
n
.
The
scalar product
of a scalar
c
∈
R
and an
n
×
m
matrix
A
over
R
, denoted
cA
,has
value
ca
i
,
j
in row
i
and column
j
.
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