Introduction to Financial Econometrics Appendix Matrix Algebra Review

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1 Introduction to Financial Econometrics Appendix Matrix Algebra Review Eric Zivot Department of Economics University of Washington January, 2000 This version: February 6, 200 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 columnsisillustratedbelow A (n m) a a 2 a m a 2 a 22 a 2m a n a n2 a nm where a ij denotes the i th row and j th column element of A A vector is simply a matrix with column For example, x x 2 x (n ) is an n vector with elements x,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 ) x n, A 0 ( 2)

2 x ( ) 2, x 0 ( ) 2 A symmetric matrix A is such that A A 0 Obviously this can only occur if A is a square matrix; ie, the number of rows of A is equal to the number of columns Forexample,considerthe2 2 matrix A 2 2 Clearly, A 0 A 2 2 Basic Matrix Operations 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, B A + B A B , 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 2and A 0 5 c A 2 2 ( ) 2 (0)

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 A (2 2) 2 4 and B (2 ) A B (2 2) (2 ) C (2 ) The resulting matrix C has 2 rows and columns In general, if A is n m and B is m p then C A B is n p As another example, let A (2 2) 2 4 and 2 B (2 ) 6 A B (2 2) (2 ) As a &nal example, let x 2, y x 0 y

4 2 The Identity Matrix The identity matrix plays a similar role as the number Multiplying any number by 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 0 I 0 denote the 2 dimensional identity matrix and let a a A 2 a 22 denote an arbitrary 2 2 matrix 0 a a I A 2 0 a 2 a 22 a a 2 A a 22 a 2 a 2 and A I a a 2 0 a 2 a 22 0 a a 2 A a 22 a 2 Inverse Matrix To be completed 4 Representing Summation Using Vector Notation Consider the sum x k x + + x k k Let x (x,,x n ) 0 be an n vector and (,,) 0 be an n vector of ones x 0 x x n x + + x k x k k 4

5 and 0 x x x n Next, consider the sum of squared x values x + + x n x 2 k x x 2 n k This sum can be conveniently represented as x 0 x x x n x x n Last, consider the sum of cross products x x 2 n x k y k x y + x n y n k This sum can be compactly represented by Note that x 0 y y 0 x x 0 y x x n y y n x y + x n y n x k k x 2 k k x k y k 5 Representing Systems of Linear Equations Using Matrix Algebra Consider the system of two linear equations k x + y () 2x y (2) which is illustrated in Figure xxx Equations () and (2) represent two straight lines which intersect at the point x 2 and y This point of intersection is determined by solving for the values of x and y such that x + y 2x y Soving for x gives x 2y Substituting this value into the equation x + y gives 2y + y and solving for y gives y / Solving for x then gives x 2/ 5

6 or The two linear equations can be written in matrix form as x 2 y where A 2 A z b x, z y and b If there was a (2 2) matrix B, with elements b ij, 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 or x y B A z B b I z B b z B b b b 2 b 2 b 22 b +b 2 b 2 +b 22 If such a matrix B exists it is called the inverse of A and is denoted A Intuitively, the inverse matrix A playsasimilarroleastheinverseofanumber Suppose a is a number; eg, a 2 we know that a a a a Similarly, in matrix algebra A A I where I is the identity matrix Next, consider solving the equation ax By simple division we have that x a x a x Similarly, in matrix algebra if we want to solve the system of equation Ax b we multiply by A and get x A b Using B A, we may express the solution for z as z A b As long as we can determine the elements in A 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 A provided the two lines are not parallel There are general numerical algorithms for &nding the elements of A 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 A Let A be a (2 2) matrix such that a a A 2 a 2 a 22 6

7 A a a 22 a 2 a 2 a22 a 2 a 2 a By brute force matrix multiplication we can verify this formula A a22 a A 2 a a 2 a a 22 a 2 a 2 a 2 a a 2 a 22 a22 a a 2 a 2 a 22 a 2 a 2 a 22 a a 22 a 2 a 2 a 2 a + a a 2 a 2 a 2 + a a 22 a22 a a 2 a 2 0 a a 22 a 2 a 2 0 a 2 a 2 + a a 22 a22 a a 2 a 2 a a 22 a 2 a a 0 2 a 2 +a a 22 a a 22 a 2 a 2 Lets apply the above rule to &nd the inverse of A in our example: A Notice that A A Our solution for z is then z A b x y 2 2 so that x 2 and y In general, if we have n linear equations in n unknown variables we may write the system of equations as a x + a 2 x a n x n b a 2 x + a 22 x a 2n x n b 2 a n x + a n2 x a nn x n b n 7

8 which we may then express in matrix form as a a 2 a n a 2 a 22 a 2n a n a n2 a nn or x x 2 x n A x b (n n) (n ) (n ) The solution to the system of equations is given by x A b where A 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 &nd the elements in A 2 Further Reading Excellent treatments of portfolio theory using matrix algebra are given in Ingersol (987), Huang and Litzenberger (988) and Campbell, Lo and MacKinlay (996) Problems To be completed References [] Campbell, JY, Lo, AW, and MacKinlay, AC (997) The Econometrics of Financial Markets Priceton, New Jersey: Princeton University Press [2] Huang, C-F, and Litzenbeger, RH (988) Foundations for Financial Economics New York: North-Holland [] Ingersoll, JE (987) Theory of Financial Decision Making Totowa, New Jersey: Rowman & Little&eld b b 2 b n 8