Introduction to Financial Econometrics Appendix Matrix Algebra Review

Eric Zivot Department of Economics University of Washington January 3, 2000 This version: February 6, 2001

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 a11 a12 . . . a1m a21 a22 . . . a2m A = . . . . ... . . . (nm) . . 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, x1 x2 x = . (n1) . . xn

is an n 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 m matrix A is a new matrix with the rows and columns of A interchanged and is denoted A0 or A| . For example, 1 4 1 2 3 , A0 = 2 5 A = 4 5 6 (32) (23) 3 6 1

x
(31)

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 2 matrix 1 2 A= . 2 1 Clearly, A =A=
0

1 = 2 , 3

(13)

x0 =

1 2 3 .

1 2 2 1

1.1
1.1.1

Basic Matrix Operations

Matrix addition and subtraction are element by element operations and only apply to matrices of the same dimension. For example, let 4 9 2 0 A= , B= . 2 1 0 7 Then A+B = AB = 1.1.2 4 9 2 1 4 9 2 1 + 2 0 0 7 2 0 0 7 = = 4+2 9+0 2+0 1+7 42 90 20 17 = = 6 9 2 8 ,

2 9 2 6

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 3 1 A= . 0 5 Then cA= 2 3 2 (1) 2 (0) 25 = 6 2 0 10 .

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 m n and B is m p then A and B are conformable. The mechanics of matrix multiplication is best explained by example. Let 1 2 1 2 1 and B = . A = 3 4 3 4 2 (22) (23) Then A B = 1 2 3 4 1 2 1 3 4 2

(22)

(23)

11+23 12+24 11+22 = 31+43 32+44 31+42 7 10 5 = = C 15 22 11

The resulting matrix C has 2 rows and 3 columns. In general, if A is n m and B is m p then C = A B is n p. As another example, let 1 2 2 A = 3 4 and B = 6 . (22) (21) Then A B 5 = 6 15+26 = 35+46 17 = . 39 1 2 3 4

(22)

(21)

As a & example, let nal 1 4 2 , y = 5 . x= 3 6

Then

4 x0 y = 1 2 3 5 = 1 4 + 2 5 + 3 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 1 0 I= 0 1 denote the 2 dimensional identity matrix and let a11 a12 A= a21 a22 denote an arbitrary 2 2 matrix. Then 1 0 a11 a12 IA = a21 a22 0 1 a11 a12 = =A a21 a22 and AI = = a11 a12 a21 a22 a11 a12 a21 a22 1 0 0 1 = A.

1.3

Inverse Matrix

To be completed.

1.4

Representing Summation Using Vector Notation

Consider the sum xk = x1 + + xk.

Let x = (x1 , . . . , xn )0 be an n 1 vector and 1 = (1, . . . , 1)0 be an n 1 vector of ones. Then 1 n X . 0 . = x1 + + xk = x 1 = x1 . . . xn . xk k=1 1 4

and

Next, consider the sum of squared x values

x1 n X . 0 1 x = 1 . . . 1 . = x1 + + xn = xk . . k=1 xn x2 = x2 + + x2 . k 1 n as

This sum can be conveniently represented x .1 0 x x = x1 . . . xn . . xn

Last, consider the sum of cross products

X x2 . = x2 + + x2 = 1 n k
k=1

xk yk = x1 y1 + xn yn .

This sum can be compactly represented by y1 n X . 0 . = x1 y1 + xn yn = x y = x1 . . . xn . xk yk . k=1 yn Note that x0 y = y0 x.

1.5

Representing Systems of Linear Equations Using Matrix Algebra

Consider the system of two linear equations x+y = 1 2x y = 1 (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 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.

The two linear equations can be written in matrix form as 1 1 x 1 = 2 1 y 1 or Az=b where A= 1 1 2 1 , z= x y and b = 1 1 .

If there was a (2 2) matrix B, with elements bij , such that B A = I, where I is the (2 2) identity matrix, then we could solve for the elements in z as follows. In the equation A z = b, pre-multiply both sides by B to give BAz = Bb = I z = B b = z = B b 1 1 = b11 1 + b12 1 b21 1 + b22 1

or

If such a matrix B exists it is called the inverse of A and is denoted A1 . Intuitively, the inverse matrix A1 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 a = a1 a = 1. Similarly, 1 in matrix algebra A 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 = a1 x. Similarly, in matrix algebra if we want to solve the system of equation Ax = b we multiply by A1 and get x = A1 b. Using B = A1 , we may express the solution for z as z = A1 b. As long as we can determine the elements in A1 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 A1 provided the two lines are not parallel. There are general numerical algorithms for & nding the elements of A1 and typical spreadsheet programs like Excel have these algorithms available. However, if A is a (2 2) matrix then there is a simple formula for A1 . Let A be a (2 2) matrix such that a11 a12 . A= a21 a22 6

x y

b11 b12 b21 b22

Then A
1

By brute force matrix multiplication we can verify this formula 1 a22 a12 a11 a12 1 A A = a21 a22 a11 a22 a21 a12 a21 a11 1 a22 a11 a12 a21 a22 a12 a12 a22 = a11 a22 a21 a12 a21 a11 + a11 a21 a21 a12 + a11 a22 1 a22 a11 a12 a21 0 = 0 a21 a12 + a11 a22 a11 a22 a21 a12 a22 a11 a12 a21 0 a11 a22 a21 a12 = a21 a12 +a11 a22 0 a11 a22 a21 a12 1 0 = . 0 1 Let s apply the above rule to & the inverse of A in our example: nd 1 1 1 1 1 1 3 A = = 3 1 . 2 1 2 2 1 3 3 Notice that A A= Our solution for z is then
1

1 = a11 a22 a21 a12

a22 a12 a21 a11

1 3 2 3

1 3 1 3

1 1 2 1

1 0 0 1

z = 1 b A 1 1 1 3 3 = 1 2 1 3 3 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 + + a1n xn = b1 a21 x1 + a22 x2 + + a2n xn = b2 . . = . . . . an1 x1 + an2 x2 + + ann xn = bn 7

or

which we may then express a11 a21 . . . an1

in matrix form as a12 a1n a22 a2n . . . an2 ann

x1 x2 . . . xn

b1 b2 . . . bn

A x = b.

(n1)

The solution to the system of equations is given by x = A1 b where A1 A = I and I is the (n n) identity matrix. If the number of equations is greater than two, then we generally use numerical algorithms to & the elements in nd A1 .

Further Reading

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

Problems

To be completed

References
[1] Campbell, J.Y., Lo, A.W., and MacKinlay, A.C. (1997). The Econometrics of Financial Markets. Priceton, New Jersey: Princeton University Press. [2] Huang, C.-F., and Litzenbeger, R.H. (1988). Foundations for Financial Economics. New York: North-Holland. [3] Ingersoll, J.E. (1987). Theory of Financial Decision Making. Totowa, New Jersey: Rowman & Little& eld.