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
columns is illustrated below
пЈ® пЈ№
a11 a12 . . . a1m
пЈЇ a21 a22 . . . a2m пЈє
пЈЇ пЈє
A =пЈЇ . .пЈє
.
пЈ°. . ... . пЈ»
. . .
(nГ—m)
an1 an2 . . . anm

where aij denotes the ith row and j th column element of A.
A vector is simply a matrix with 1 column. For example,
пЈ® пЈ№
x1
пЈЇ x2 пЈє
пЈЇ пЈє
x =пЈЇ . пЈє
пЈ°.пЈ».
(nГ—1)
xn
is an n Г— 1 vector with elements x1 , x2 , . . . , xn . 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 Г— m matrix A is a new matrix with the rows and columns
of A interchanged and is denoted A0 or A| . For example,
пЈ® пЈ№
В· Вё 14
123
, A0 = пЈ° 2 5 пЈ»
A=
456 (3Г—2)
(2Г—3)
36

1
пЈ® пЈ№
1 ВЈ В¤
= пЈ° 2 пЈ», x0 =
x 123.
(3Г—1) (1Г—3)
3

A symmetric matrix A is such that A = A0 . 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.
For example, consider the 2 Г— 2 matrix
В· Вё
12
A= .
21

Clearly, В· Вё
12
A0 = A = .
21

1.1 Basic Matrix Operations
Matrix addition and subtraction are element by element operations and only apply
to matrices of the same dimension. For example, let
В· Вё В· Вё
49 20
A= , B= .
21 07

Then
В· Вё В· Вё В· Вё В· Вё
49 20 4+2 9+0 69
A+B = + = = ,
21 07 2+0 1+7 28
В· Вё В· Вё В· Вё В· Вё
4в€’2 9в€’0
49 20 29
Aв€’B = в€’ = = .
2в€’0 1в€’7 2 в€’6
21 07

1.1.2 Scalar Multiplication
Here we refer to the multiplication of a matrix by a scalar number. This is also an
element-by-element operation. For example, let c = 2 and
В· Вё
3 в€’1
A= .
05

Then В· Вё В· Вё
2 В· 3 2 В· (в€’1) 6 в€’2
cВ·A= = .
2 В· (0) 2В·5 0 10

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. For
example, if A is mГ— n and B is m Г— p then A and B are conformable. The mechanics
of matrix multiplication is best explained by example. Let
В· Вё В· Вё
12 121
and B =
A= .
34 342
(2Г—2) (2Г—3)

Then
В· ВёВ· Вё
12 121
AВ·B В·
=
34 342
(2Г—2) (2Г—3)
В· Вё
1В·1+2В·3 1В·2+2В·4 1В·1+2В·2
=
3В·1+4В·3 3В·2+4В·4 3В·1+4В·2
В· Вё
7 10 5
= =C
15 22 11 (2Г—3)

The resulting matrix C has 2 rows and 3 columns. In general, if A is n Г— m and B
is m Г— p then C = A В· B is n Г— p.
As another example, let
В· Вё В·Вё
12 2
A = 3 4 and B = 6 .
(2Г—2) (2Г—1)

Then
В· ВёВ· Вё
12 5
AВ·B В·
=
34 6
(2Г—2) (2Г—1)
В· Вё
1В·5+2В·6
=
3В·5+4В·6
В· Вё
17
= .
39

As a & example, let
nal
пЈ® пЈ№ пЈ® пЈ№
1 4
x = пЈ° 2 пЈ», y = пЈ° 5 пЈ».
3 6

Then пЈ®
пЈ№
4
ВЈ В¤
x0 y = 1 2 3 В· пЈ° 5 пЈ» = 1 В· 4 + 2 В· 5 + 3 В· 6 = 32
6

3
1.2 The Identity Matrix
The identity matrix plays a similar role as the number 1. Multiplying any number by
1 gives back that number. In matrix algebra, pre or post multiplying a matrix A by
a conformable identity matrix gives back the matrix A. To illustrate, let
В· Вё
10
I=
01

denote the 2 dimensional identity matrix and let
В· Вё
a11 a12
A=
a21 a22

denote an arbitrary 2 Г— 2 matrix. Then
В· ВёВ· Вё
10 a11 a12
IВ·A = В·
01 a21 a22
В· Вё
a11 a12
= =A
a21 a22

and
В· ВёВ· Вё
a11 a12 10
AВ·I = В·
a21 a22 01
В· Вё
a11 a12
= = A.
a21 a22

1.3 Inverse Matrix
To be completed.

1.4 Representing Summation Using Vector Notation
Consider the sum
n
X
xk = x1 + В· В· В· + xk.
k=1

Let x = (x1 , . . . , xn )0 be an n Г— 1 vector and 1 = (1, . . . , 1)0 be an n Г— 1 vector of
ones. Then
пЈ®пЈ№
1 n
X
ВЈ В¤ пЈЇ.пЈє
0
x 1 = x1 . . . xn В· пЈ° . пЈ» = x1 + В· В· В· + xk = xk
.
1 k=1

4
and пЈ®
пЈ№
x1 n
X
ВЈ В¤пЈЇ.пЈє
0
1 x = 1 . . . 1 В· пЈ° . пЈ» = x1 + В· В· В· + xn = xk .
.
xn k=1

Next, consider the sum of squared x values
n
X
x2 = x2 + В· В· В· + x2 .
k 1 n
k=1

This sum can be conveniently represented as
пЈ® пЈ№
x
В¤ пЈЇ .1 n
X
ВЈ пЈє
x0 x = x1 . . . xn В· пЈ° . 2 2
x2 .
пЈ» = x1 + В· В· В· + xn =
. k
xn k=1

Last, consider the sum of cross products
n
X
xk yk = x1 y1 + В· В· В· xn yn .
k=1

This sum can be compactly represented by
пЈ® пЈ№
y
В¤ пЈЇ .1 пЈє Xn
ВЈ
0
x y = x1 . . . xn В· пЈ° . пЈ» = x1 y1 + В· В· В· xn yn = xk yk .
.
yn k=1

Note that x0 y = y0 x.

1.5 Representing Systems of Linear Equations Using Matrix
Algebra
Consider the system of two linear equations

(1)
x+y = 1
2x в€’ y = 1 (2)

which is illustrated in Figure xxx. Equations (1) and (2) represent two straight lines
which intersect at the point x = 2 and y = 1 . This point of intersection is determined
3 3
by solving for the values of x and y such that x + y = 2x в€’ y 1 .
1
Soving for x gives x = 2y. Substituting this value into the equation x + y = 1 gives 2y + y = 1
and solving for y gives y = 1/3. Solving for x then gives x = 2/3.

5
The two linear equations can be written in matrix form as
В· ВёВ· Вё В· Вё
11 x 1
=
2 в€’1 y 1
or
AВ·z=b
where В· Вё В· Вё В· Вё
11 x 1
and b =
A= , z= .
2 в€’1 y 1
If there was a (2 Г— 2) matrix B, with elements bij , such that B В· A = I, where I
is the (2 Г— 2) identity matrix, then we could solve for the elements in z as follows. In
the equation A В· z = b, pre-multiply both sides by B to give

BВ·AВ·z = BВ·b
=в‡’ I В· z = B В· b
=в‡’ z = B В· b

or В· Вё В· ВёВ· Вё В· Вё
b11 В· 1 + b12 В· 1
x b11 b12 1
= =
b21 В· 1 + b22 В· 1
y b21 b22 1
If such a matrix B exists it is called the inverse of A and is denoted Aв€’1 . In-
tuitively, the inverse matrix Aв€’1 plays a similar role as the inverse of a number.
1
Suppose a is a number; e.g., a = 2. Then we know that a В· a = aв€’1 a = 1. Similarly,
in matrix algebra Aв€’1 A = I where I is the identity matrix. Next, consider solving
1
the equation ax = 1. By simple division we have that x = a x = aв€’1 x. Similarly, in
matrix algebra if we want to solve the system of equation Ax = b we multiply by
Aв€’1 and get x = Aв€’1 b.
Using B = Aв€’1 , we may express the solution for z as

z = Aв€’1 b.

As long as we can determine the elements in Aв€’1 then we can solve for the values of
x and y in the vector z. Since the system of linear equations has a solution as long as
the two lines intersect, we can determine the elements in Aв€’1 provided the two lines
are not parallel.
nding the elements of Aв€’1 and typical
There are general numerical algorithms for &
spreadsheet programs like Excel have these algorithms available. However, if A is a
(2 Г— 2) matrix then there is a simple formula for Aв€’1 . Let A be a (2 Г— 2) matrix such
that В· Вё
a11 a12
A= .
a21 a22

6
Then В· Вё
1 a22 в€’a12
Aв€’1 = .
в€’a21 a11
a11 a22 в€’ a21 a12
By brute force matrix multiplication we can verify this formula
В· ВёВ· Вё
1 a22 в€’a12 a11 a12
Aв€’1 A =
a11 a22 в€’ a21 a12 в€’a21 a11 a21 a22
В· Вё
1 a22 a11 в€’ a12 a21 a22 a12 в€’ a12 a22
=
a11 a22 в€’ a21 a12 в€’a21 a11 + a11 a21 в€’a21 a12 + a11 a22
В· Вё
1 a22 a11 в€’ a12 a21 0
=
в€’a21 a12 + a11 a22
0
a11 a22 в€’ a21 a12
В· a22 a11 в€’a12 a21 Вё
0
a11 a22 в€’a21 a12
= в€’a21 a12 +a11 a22
0 a11 a22 в€’a21 a12
В· Вё
10
= .
01

Let s apply the above rule to & the inverse of A in our example:
nd
В· Вё В·1 1 Вё
1 в€’1 в€’1
Aв€’1 = = 3 в€’1 . 3
2
в€’1 в€’ 2 в€’2 1 3 3

Notice that В· ВёВ· Вё В· Вё
1 1
11 10
Aв€’1 A = 3 3 = .
2 в€’1
2 в€’1 01
3 3
Our solution for z is then

z = В· в€’1 b Вё В· Вё
A
1 1
1
3 3
= 2 в€’1
1
3
Вё3 В· Вё
В·2
x
3
= =
1
y
3

so that x = 2 and y = 1 .
3 3
In general, if we have n linear equations in n unknown variables we may write the
system of equations as

a11 x1 + a12 x2 + В· В· В· + a1n xn = b1
a21 x1 + a22 x2 + В· В· В· + a2n xn = b2
.
.=. .
. .
an1 x1 + an2 x2 + В· В· В· + ann xn = bn

7
which we may then express in matrix form as
пЈ® пЈ№пЈ® пЈ№ пЈ® пЈ№
a12 В· В· В· a1n
a11 x1 b1
пЈЇ a21 a22 В· В· В· a2n пЈє пЈЇ пЈєпЈЇ пЈє
x2 b2
пЈЇ пЈєпЈЇ пЈєпЈЇ пЈє
пЈЇ. . пЈєпЈЇ пЈє=пЈЇ пЈє
. .
пЈ°. . пЈ»пЈ° . .
пЈ»пЈ° пЈ»
. . . .
an2 В· В· В· ann
an1 xn bn
or
A В· x = b.
(nГ—1) (nГ—1)
(nГ—n)

The solution to the system of equations is given by

x = Aв€’1 b

where Aв€’1 A = I and I is the (n Г— n) identity matrix. If the number of equations is
greater than two, then we generally use numerical algorithms to & the elements in
nd
Aв€’1 .

Excellent treatments of portfolio theory using matrix algebra are given in Ingersol
(1987), Huang and Litzenberger (1988) and Campbell, Lo and MacKinlay (1996).

3 Problems
To be completed

References
 Campbell, J.Y., Lo, A.W., and MacKinlay, A.C. (1997). The Econometrics of
Financial Markets. Priceton, New Jersey: Princeton University Press.

 Huang, C.-F., and Litzenbeger, R.H. (1988). Foundations for Financial Eco-
nomics. New York: North-Holland.

 Ingersoll, J.E. (1987). Theory of Financial Decision Making. Totowa, New Jersey:
Rowman & Little& eld.

8