10
Elements of Matrix Analysis
CONTENTS
10.1 Review of Elementary Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10.1.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
10.2 Projection Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
10.3 Idempotent Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
10.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
10.5 Kronecker Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
10.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
10.7 Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
10.8 Numerical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Many of the traditional econometric applications involve the estimation of
linear equations or systems of equations to describe the behavior of individuals
or groups of individuals. For example, we can specify that the quantity of an
input demanded by a firm is a linear function of the firm’s output price, the
price of the input, and the price of other inputs
x
D
t
= α
0
+ α
1
p
t
+ α
2
w
1t
+ α
2
w
2t
+ α
3
w
3t
+
t
(10.1)
where x
D
t
is the quantity of the input demanded at time t, p
t
is the price of
the firm’s output, w
1t
is the price of the input, w
2t
and w
3t
are the prices
of other inputs used by the firm, and
t
is a random error. Under a variety
of assumptions such as those discussed in Chapter 6 or by assuming that
t
∼ N
0, σ
2
, Equation 10.1 can be estimated using matrix methods. For
example, assume that we have a simple linear specification
y
i
= α
0
+ α
1
x
i
+
i
. (10.2)
Section 7.6 depicts the derivation of two sets of normal equations.
−
1
N
N
X
i=1
y
i
+ α
0
+ α
1
1
N
N
X
i=1
x
i
= 0
−
1
N
N
X
i=1
x
i
y
i
+ α
0
1
N
N
X
i=1
x
i
+ α
1
1
N
N
X
i=1
x
2
i
= 0.
(10.3)
217
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