Appendix A
Matrix Algebra
1.
For the matrices
A
=
and
B
=
compute
AB
,
A
′
B
′
, and
BA
.
24
15
62
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
133
241
⎡
⎣
⎢
⎤
⎦
⎥
AB
=
,
BA
=
,
A
′
B
′
=
(
BA
)
′
=
.
23 25
14
30
⎡
⎣
⎢
⎤
⎦
⎥
10 22
10
11
23
8
10
26
20
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
10
11 10
22
23 26
10
8
20
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2.
Prove that
tr
(
AB
) =
tr
(
BA
) where
A
and
B
are any two matrices that are conformable for both
multiplications.
They need not be square.
The
i
th diagonal element of
AB
is
.
Summing over
i
produces
tr
(
AB
) =
.
The jth diagonal element of
BA
is
.
Summing over
i
produces
tr
(
BA
) =
.
a
ij
b
ji
j
∑
a
ij
b
ji
i
i
∑
∑
b
ji
a
ij
j
∑
b
ji
a
ij
j
i
∑
∑
3.
Prove that
tr
(
A
′
A
) =
.
a
ij
j
i
2
∑
∑
The
j
th diagonal element of
A
′
A
is the inner product of the
j
th column of
A
, or
Summing
over
j
produces
tr
(
A
′
A
)
=
.
a
ij
i
2
.
∑
a
ij
i
j
a
ij
j
i
22
∑
∑
=
∑
∑
4.
Expand the matrix product
X
= {[
AB
+ (
CD
)
′
][(
EF
)
1
+
GH
]}
′
.
Assume that all matrices are square and
E
and
F
are nonsingular.
In parts, (
CD
)
′
=
D
′
C
′
and (
EF
)
1
=
F
1
E
1
.
Then, the product is
{[
AB
+ (
CD
)
′
][(
EF
)
1
+
GH
]}
′
=
(
ABF
1
E
1
+
ABGH
+
D
′
C
′
F
1
E
1
+
D
′
C
′
GH
)
′
=
(
E
1
)
′
(
F
1
)
′
B
′
A
′
+
H
′
G
′
B
′
A
′
+ (
E
1
)
′
(
F
1
)
′
CD
+
H
′
G
′
CD
.
±
±
5.
Prove for that for
K
×
1 column vectors,
x
i
i
= 1,.
..,
n
, and some nonzero vector,
a
,
()
1
''
n
ii
i
n
=
′
−−
=
+
−
−
∑
0
xaxa X
M
X xaxa
.
Write
x
i

a
as [(
x

i
x
) + (
x

a
)].
Then, the sum is
i
n
=
∑
1
[(
x
i

x
) + (
x

a
)] [(
x
i

x
) + (
x

a
)]
′
=
(
x
i

i
n
=
∑
1
x
)(
x
i

x
)
′
+
(
i
n
=
∑
1
x

a
) (
x

a
)
′
+
(
x
i

i
n
=
∑
1
x
)(
x

a
)
′
+
(
i
n
=
∑
1
x

a
) (
x
i

x
)
′
Since (
x

a
) is a vector of constants, it may be moved out of the summations. Thus, the fourth term is
(
x

a
)
(
x
i

i
n
=
∑
1
x
)
′
=
0.
The third term is likewise.
The first term is
X
′
M
0
X
by the definition while the
second is
n
(
x

a
) (
x

a
)
′
.
±
6.
Let
A
be any square matrix whose columns are [
a
1
,
a
2
,...,
a
M
] and let
B
be any rearrangement of the columns
of the
M
×
M
identity matrix.
What operation is performed by the multiplication
AB
?
What about
BA
?
B
is called a permutation matrix.
Each column of
B
, say,
b
i
, is a column of an identity matrix.
The
j
th column of the matrix product
AB
is
A b
i
which is the
j
th column of
A
.
Therefore, post multiplication of
A
by
B
simply rearranges (permutes) the columns of
A
(hence the name).
Each row of the product
BA
is one of
the rows of
A
, so the product
BA
is a rearrangement of the rows of
A
.
Of course,
A
need not be square for us
155