Databases Reference
In-Depth Information
If a matrix consists of a single column
(
N
=
1
)
, it is called a
column matrix
or
vector
of
dimension
M
. If it consists of a single row
(
M
=
1
)
, it is called a
row matrix
or
vector
of
dimension
N
.
The
transpose
A
T
of a matrix
A
is the
N
×
M
matrix obtained by writing the rows of the
matrix as columns and the columns as rows:
⎡
⎤
a
00
a
10
···
a
(
M
−
1
)
0
⎣
⎦
a
01
a
11
···
a
(
M
−
1
)
1
A
T
=
(B.3)
.
.
.
a
0
(
N
−
1
)
a
1
(
N
−
1
)
···
a
M
−
1
N
−
1
The transpose of a column matrix is a row matrix and vice versa.
Two matrices
A
and
B
are said to be equal if they are of the same order and their corre-
sponding elements are equal; that is,
A
=
B
⇔
a
ij
=
b
ij
,
i
=
0
,
1
,...
M
−
1
;
j
=
0
,
1
,...
N
−
1
(B.4)
B.2 Matrix Operations
You can add, subtract, and multiply matrices, but since matrices come in all shapes and sizes,
there are some restrictions as to what operations you can perform with what kind of matrices.
In order to add or subtract two matrices, their dimensions have to be identical—same number
of rows and same number of columns. In order to multiply two matrices, the order in which
they are multiplied is important. In general
A
A
. Multiplication is only
defined for the case where the number of columns of the first matrix is equal to the number of
rows of the second matrix. The reasons for these restrictions will become apparent when we
look at how the operations are defined.
When we add two matrices, the resultant matrix consists of elements that are the sum of
the corresponding entries in the matrices being added. Let us add two matrices
A
and
B
where
×
B
is not equal to
B
×
a
00
a
01
a
02
a
10
a
11
a
12
A
=
and
b
00
b
01
b
02
b
10
b
11
b
12
B
=
The sum of the two matrices,
C
, is given by
c
00
c
12
c
13
c
21
c
22
c
23
a
00
+
b
00
a
01
+
b
01
a
02
+
b
02
C
=
=
(B.5)
a
10
+
b
10
a
11
+
b
11
a
12
+
b
12
Notice that each element of the resulting matrix
C
is the sum of corresponding elements of the
matrices
A
and
B
. In order for the two matrices to have corresponding elements, the dimension
of the two matrices has to be the same. Therefore, addition is only defined for matrices with
