Problem Set 11 due November 29 MATH 110 Linear Algebra Each problem is worth 10 points PART 1 1 2 3 4 Curtis Curtis Curtis Curtis p p p p 276 276 276 288 2 4 9 4 PART 2 Problem 1 20 Show that for any 1 n row matrix X and any n n matrix A we have that XAX t XBX t where B is the symmetric matrix B 12 A At Problem 2 10 Let T V V be a symmetric transformation on a real Euclidean space V with quadratic form Q x T x x Prove that if Q does not change sign sign on V i e Q x 0 x V or Q x 0 x V then Q x 0 implies that T x 0 Hint consider the function p t Q x ty and show that p0 0 0 Problem 3 10 a Prove that the set of symmetric matrices forms a subspace of dimension n n 1 in the vector space of real n n matrices which has dimension n2 2 b Prove that the set of skew symmetric matrices forms a subspace of dimension n n 1 in the vector space of real n n matrices which has dimension 2 n2 Problem 4 10 Let T be a symmetric linear transformation on a real Euclidean space Prove that T has a square root i e there is a symmetric linear transformation S such that T S 2 Problem 5 10 Define the index of a real symmetric matrix A to be the number of strictly positive eigenvalues of A minus the number of strictly negative eigenvalues 1 of A Let QA x and QB x be the quadratic forms associated with A and B and suppose that QA x QB x for all vectors x Prove that the index of A is less than or equal to the index of B Optional Problem Let T V V be a symmetric linear transformation on a real Euclidean space V and let Q x T x x Assume that there is an extremum maximum or minimum at u for Q x among all the values that Q takes on the unit sphere i e Q u is either a max or a min amongst all the values Q x with x x 1 Show that u is an eigenvector for T 2
View Full Document
Unlocking...