Databases Reference
In-Depth Information
by
AB
−
1
. Once we have the definition of an inverse of a matrix, the rules of multiplication
apply.
So how do we define the inverse of a matrix? Following the definition for real numbers,
in order to define the inverse of a matrix we need to have the matrix counterpart of 1. In
matrices this counterpart is called the
identity matrix
. The identity matrix is a square matrix
with diagonal elements being 1 and off-diagonal elements being 0. For example, a 3
×
3
identity matrix is given by
⎡
⎣
⎤
⎦
100
010
001
=
(B.7)
I
The identitymatrix behaves like the number in the matrix world. If we multiply anymatrix with
the identity matrix (of appropriate dimension), we get the original matrix back. Given a square
matrix
A
, we define its inverse,
A
−
1
, as the matrix that when premultiplied or postmultiplied
by
A
results in the identity matrix. For example, consider the matrix
34
12
A
=
(B.8)
The inverse matrix is given by
1
−
2
A
−
1
=
(B.9)
−
0
.
51
.
5
To check that this is indeed the inverse matrix, let us multiply them:
34
12
1
10
01
−
2
=
(B.10)
−
0
.
51
.
5
and
1
34
12
10
01
−
2
=
(B.11)
−
0
.
51
.
5
If
A
is a vector of dimension
M
, we can define two specific kinds of products. If
A
is a
column matrix, then the
inner product
or
dot product
is defined as
M
−
1
A
T
A
a
i
0
=
(B.12)
i
=
0
and the
outer product
or
cross product
is defined as
⎡
⎤
a
00
a
00
a
00
a
10
···
a
00
a
(
M
−
1
)
0
⎣
⎦
a
10
a
00
a
10
a
10
···
a
10
a
(
M
−
1
)
0
AA
T
=
(B.13)
.
.
.
···
a
(
M
−
1
)
0
a
00
a
{
(
M
−
1
)
1
}
a
10
a
(
M
−
1
)
0
a
(
M
−
1
)
0
Notice that the inner product results in a scalar, while the outer product results in a matrix.
