This preview shows pages 1–4. Sign up to view the full content.
Introduction to Financial Econometrics
Appendix Matrix Algebra Review
Eric Zivot
Department of Economics
University of Washington
January 3, 2000
This version: February 6, 2001
1 Matrix Algebra Review
A
matrix
is just an array of numbers. The
dimension
of a matrix is determined by
the number of its rows and columns. For example, a matrix
A
with
n
rows and
m
co
lumnsisi
l
lustratedbe
low
A
(
n
×
m
)
=
a
11
a
12
... a
1
m
a
21
a
22
2
m
.
.
.
.
.
.
...
.
.
.
a
n
1
a
n
2
nm
where
a
ij
denotes the
i
th
row and
j
th
column element of
A
.
A
vector
is simply a matrix with
1
column. For example,
x
(
n
×
1)
=
x
1
x
2
.
.
.
x
n
is an
n
×
1
vector with elements
x
1
,x
2
,...,x
n
.
Vectors and matrices are often written
in bold type (or underlined) to distinguish them from scalars (single elements of
vectors or matrices).
The
transpose
of an
n
×
m
matrix
A
is a new matrix with the rows and columns
of
A
interchanged and is denoted
A
0
or
A

.
For example,
A
(2
×
3)
=
·
123
456
¸
,
A
0
(3
×
2)
=
14
25
36
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document x
(3
×
1)
=
1
2
3
,
x
0
(1
×
3)
=
£
123
¤
.
A
symmetric
matrix
A
is such that
A
=
A
0
.
Obviously this can only occur if
A
is a
square
matrix; i.e., the number of rows of
A
is equal to the number of columns.
Forexamp
le
,cons
iderthe
2
×
2
matrix
A
=
·
12
21
¸
.
Clearly,
A
0
=
A
=
·
¸
.
1.1 Basic Matrix Operations
1.1.1 Addition and subtraction
Matrix addition and subtraction are element by element operations and only apply
to matrices of the same dimension. For example, let
A
=
·
49
¸
,
B
=
·
20
07
¸
.
Then
A
+
B
=
·
¸
+
·
¸
=
·
4+2 9+0
2+0 1+7
¸
=
·
69
28
¸
,
A
−
B
=
·
¸
−
·
¸
=
·
4
−
29
−
0
2
−
01
−
7
¸
=
·
2
−
6
¸
.
1.1.2 Scalar Multiplication
Here we refer to the multiplication of a matrix by a scalar number. This is also an
elementbyelement operation. For example, let
c
=2
and
A
=
·
3
−
1
05
¸
.
Then
c
·
A
=
·
2
·
32
·
(
−
1)
2
·
(0)
2
·
5
¸
=
·
6
−
2
0
¸
.
2
1.1.3 Matrix Multiplication
Matrix multiplication only applies to
conformable
matrices.
A
and
B
are conformable
matrices of the number of columns in
A
is equal to the number of rows in
B
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 09/13/2011 for the course ECON 503 taught by Professor Pujara during the Spring '11 term at Punjab Engineering College.
 Spring '11
 Pujara
 Econometrics

Click to edit the document details