Fin500J Mathematical Foundations in Finance Topic 1: Matrix Algebra Philip H. Dybvig Reference: Mathematics for Economists, Carl Simon and Lawrence Blume, Chapter 8 Chapter 9 and Chapter 16 Slides designed by Yajun Wang 1 Fall 2010 Olin Business School Fin500J Topic 1

Outline Definition of a Matrix Operations of Matrices Determinants Inverse of a Matrix Linear System Matrix Definiteness Fall 2010 Olin Business School2Fin500J Topic 1

Matrix (Basic Definitions) An k × n matrix A is a rectangular array of numbers with k rows and n columns. (Rows are horizontal and columns are vertical.) The numbers k and n are the dimensions of A. The numbers in the matrix are called its entries. The entry in row i and column j is called a ij. 3Fall 2010 Olin Business SchoolFin500J Topic 1

Operations with Matrices (Sum, Difference) Sum, Difference If A and B have the same dimensions, then their sum, A + B, is obtained by adding corresponding entries. In symbols, (A + B) ij = a ij + b ij. If A and B have the same dimensions, then their difference, A − B, is obtained by subtracting corresponding entries. In symbols, (A - B) ij = a ij - b ij. Example: 4Fall 2010 Olin Business SchoolFin500J Topic 1

Operations with Matrices (Scalar Multiple) Scalar Multiple If A is a matrix and r is a number (sometimes called a scalar in this context), then the scalar multiple, rA, is obtained by multiplying every entry in A by r. In symbols, (rA) ij = ra ij. Example: 5Fall 2010 Olin Business SchoolFin500J Topic 1

Operations with Matrices (Product) Product If A has dimensions k × m and B has dimensions m × n, then the product AB is defined, and has dimensions k × n. The entry (AB) ij is obtained by multiplying row i of A by column j of B, which is done by multiplying corresponding entries together and then adding the results i.e., 6Fall 2010 Olin Business SchoolFin500J Topic 1

Laws of Matrix Algebra The matrix addition, subtraction, scalar multiplication and matrix multiplication, have the following properties. Fall 2010 Olin Business School7Fin500J Topic 1

Operations with Matrices (Transpose) Transpose The transpose, A T, of a matrix A is the matrix obtained from A by writing its rows as columns. If A is an k×n matrix and B = A T then B is the n×k matrix with b ij = a ji. If A T =A, then A is symmetric. Example: 8Fall 2010 Olin Business SchoolFin500J Topic 1

Determinants Determinant is a scalar Defined for a square matrix Is the sum of selected products of the elements of the matrix, each product being multiplied by +1 or -1 9Fall 2010 Olin Business School M ij =det(A ij ), A ij is the (n-1)×(n-1) submatrix obtained by deleting row i and column j from A. Fin500J Topic 1

Determinants The determinant of a 3 ×3 matrix is Example 10 The determinant of a 2 ×2 matrix A is In Matlab: det(A) = det(A) Fall 2010 Olin Business SchoolFin500J Topic 1

Inverse of a Matrix Definition. If A is a square matrix, i.e., A has dimensions n×n. Matrix A is nonsingular or invertible if there exists a matrix B such that AB=BA=I n. For example.  Common notation for the inverse of a matrix A is A -1  If A is an invertible matrix, then (A T ) -1 = (A -1 ) T  The inverse matrix A -1 is unique when it exists.  If A is invertible, A -1 is also invertible  A is the inverse matrix of A -1. (A -1 ) -1 =A. In Matlab: A -1 = inv(A) Matrix division: A/B = AB -1 11Fall 2010 Olin Business SchoolFin500J Topic 1

Calculation of Inversion using Determinants Def: For any n×n matrix A, let C ij denote the (i,j) th cofactor of A, that is, (-1) i+j times the determinant of the submatrix obtained by deleting row i and column j form A, i.e., C ij = (-1) i+j M ij. The n×n matrix whose (i,j)th entry is C ji, the (j,i)th cofactor of A is called the adjoint of A and is written adj A. thus 12 Fall 2010 Olin Business SchoolFin500J Topic 1

Calculation of Inversion using Determinants thus Example: find the inverse of the matrix Solve: 13 Using Determinants to find the inverse of a matrix can be very complicated. Gaussian elimination is more efficient for high dimension matrix. 13Fall 2010 Olin Business SchoolFin500J Topic 1

Calculation of Inversion using Gaussian Elimination 14 Elementary row operations: o Interchange two rows of a matrix o Change a row by adding to it a multiple of another row o Multiply each element in a row by the same nonzero number To calculate the inverse of matrix A, we apply the elementary row operations on the augmented matrix [A I] and reduce this matrix to the form of [I B] The right half of this augmented matrix B is the inverse of A 14 Fall 2010 Olin Business School Fin500J Topic 1

Calculation of inversion using Gaussian elimination I is the identity matrix, and use Gaussian elimination to obtain a matrix of the form The matrix 15Fall 2010 Olin Business School is then the matrix inverse of A Fin500J Topic 1

Example The matrix 16 is then the matrix inverse of A (ii)+(-12) ×(i), (iii)+(-3) ×(i), (iii)+(ii) ×(1/10) 16 Fall 2010 Olin Business School Fin500J Topic 1

The system of linear equations 17Fall 2010 Olin Business School Systems of Equations in Matrix Form can be rewritten as the matrix equation Ax=b, where If an n ×n matrix A is invertible, then it is nonsingular, and the unique solution to the system of linear equations Ax=b is x=A -1 b. Fin500J Topic 1

Example: solve the linear system 18Fall 2010 Olin Business School In Matlab >>x=inv(A)*b or >> x=A\b b Fin500J Topic 1

19Fall 2010 Olin Business School Matrix Operations in Matlab >> A=[2 3; 1 1; 1 0] A = >> B1=[1 1; 0 1; 2 4] B1 = >> B2=[1 1 1; 1 0 2] B2 = >> A=[2 3; 1 1; 1 0] A = >> B1=[1 1; 0 1; 2 4] B1 = >> B2=[1 1 1; 1 0 2] B2 = >> A+B1 ans = >> A-B1 ans = >> A*B2 ans = >> A+B1 ans = >> A-B1 ans = >> A*B2 ans = Sum Difference Product Fin500J Topic 1

20Fall 2010 Olin Business School Matrix Operations in Matlab >> C=[1 1 1; ; 3 4 1] C = >> C=[1 1 1; ; 3 4 1] C = >> C' ans = >> det(C) ans = 35 >> inv(C) ans = >> C' ans = >> det(C) ans = 35 >> inv(C) ans = transpose determinant inverse Fin500J Topic 1

Positive Definite Matrix Fall 2010 Olin Business School21Fin500J Topic 1

Negative Definite, Positive Semidefinite, Negative Semidefinite, Indefinite Matrix Fall 2010 Olin Business School22 Let A be an N×N symmetric matrix, then A is negative definite if and only if v T Av <0 for all v≠0 in R N positive semidefinite if and only if v T Av ≥0 for all v≠0, in R N negative semidefinite if and only if v T Av ≤0 for all v≠0, in R N indefinite if and only if v T Av >0 for some v in R N and <0 for other v in R N Fin500J Topic 1