UNIVERSITY OF WISCONSIN MADISON Computer Sciences Department CS513 Spring 99 Prof Ron Midterm Exam My name is Answer every question below Write your answer into your blue book Use a new page for each question You are not allowed to use books or notes A calculator of any type is permitted provided that it is not preprogrammed with code relevant to cs513 Be brief and to the point with your answers Yes no answers carry no credit unless reasoning is provided Turn in your exam sheet together with your blue book 1 30 points 5 15 10 You are given the following 2 2 matrix A 2 1 A 3 0 a Find the spectrum of this matrix b Find the 2 and 1 norm of this matrix c Find the 2 condition number of A if possible without computing the inverse of A State any result from class that you use here 2 25 points 10 15 a Let v be the column vector defined by v 0 3 2 4 4 Find a Householder matrix H such that Hv 0 3 with a some number that you might choose b QR factor the matrix 5 0 0 0 a 0 0 to suit your needs 3 2 4 4 1 0 0 0 3 20 points 10 10 a Describe briefly an efficient algorithm for computing the determinant of a square n n matrix What is the complexity of your algorithm b Use a in order to find the determinant of the matrix 4 2 2 A 2 2 2 2 2 3 4 20 points 5 10 5 a A is a square matrix Define the notion A is positive definite b State two conditions each of which is equivalent for a symmetric matrix A to the positive definiteness of that matrix c Check whether the matrix in the previous problem 3 b is positive definite 5 25 points 5 20 points Let V be the linear span of the vectors 1 1 1 1 0 0 You are asked to find a vector v in V which is as close as possible to the vector 1 u 2 4 a Define rigorously the notion of as close as possible If there are several possible definitions choose the one that will allow you to solve b below b Find that closest vector v