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product is not the zero vector. A set of rows of a matrix
A
over a field
R
is
linearly indepen-
dent
if all linear combinations are non-zero except when each scalar is zero.
The
rank of an
n
,
f
(
n
)
n
2
×
m
matrix
A
over a field
R
rank
:
R
→
, is the maximum
number of linearly independent rows of
A
. It is also the maximum number of linearly inde-
pendent columns of
A
. (See Problem
6.5
.) We write
rank(
A
)=
f
(
n
)
rank
(
A
)
.An
n
×
n
matrix
A
is
non-singular
if
rank(
A
)=
n
.
If an
n
is non-singular, it has an
inverse
A
−
1
that is an
n
×
n
matrix
A
over a field
R
×
n
matrix with the following properties:
AA
−
1
=
A
−
1
A
=
I
n
n
identity matrix. That is, there is a (partial) inverse function
f
(
n
)
inv
where
I
n
is the
n
×
:
that is defined for non-singular square matrices
A
such that
f
(
n
)
n
2
n
2
R
→R
inv
(
A
)=
A
−
1
.
f
(
n
)
inv
is partial because it is not defined for singular matrices. Below we exhibit a matrix and its
inverse over a field
R
.
11
−
−
1
=
1
−
1
11
11
Algorithms for matrix inversion are given in Section
6.5
.
We now show that the inverse
(
AB
)
−
1
of the product
AB
of two invertible matrices,
A
and
B
,overafield
R
is the product of their inverses in reverse order.
LEMMA
6.2.1
Let
A
and
B
be invertible square matrices over a field
R
. Then the following
relationship holds:
(
AB
)
−
1
=
B
−
1
A
−
1
Proof
To s h ow t h a t
(
AB
)
−
1
=
B
−
1
A
−
1
, we multiply
AB
either on the left or right by
B
−
1
A
−
1
to produce the identity matrix:
AB
(
AB
)
−
1
ABB
−
1
A
−
1
A
(
BB
−
1
)
A
−
1
AA
−
1
=
=
=
=
I
(
AB
)
−
1
AB
B
−
1
A
−
1
AB
B
−
1
(
A
−
1
A
)
B
B
−
1
B
=
=
=
=
I
The transpose of the product of an
m
×
n
matrix
A
and an
n
×
p
matrix
B
over a ring
R
is the product of their transposes in reverse order:
(
AB
)
T
=
B
T
A
T
(See Problem
6.6
.) In particular, the following identity holds for an
m
×
n
matrix
A
and a
column
n
-vector
x
:
=(
A
x
)
T
x
T
A
T
A
block matrix
is a matrix in which each entry is a matrix with fixed dimensions. For
example, when
n
is even it may be convenient to view an
n
×
n
matrix as a 2
×
2matrixwhose
four entries are
(
n/
2
)
(
n/
2
)
matrices.
Two special types of matrix that are frequently encountered are the Toeplitz and circulant
matrices. An
n
×
×
n
Toeplitz matrix
T
has the property that its
(
i
,
j
)
entry
t
i
,
j
=
a
r
for
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