Math 545 Linear Algebra with Applications Spring 2013 Homework Set 4 Due Wedesday, 17 April 1. Suppose A 2 Cnn is skew-Hermitian, so A= ??A. Prove that the eigenvalues of A are pure imaginary, so =...

Math 545 Linear Algebra with Applications Spring 2013 Homework Set 4 Due Wedesday, 17 April nn  1. Suppose A2 C is skew-Hermitian, so A =A. Prove that the eigenvalues  of A are pure imaginary, so  =. nn T n 2. Suppose A2R and consider the quadratic form q(x) =x Ax. Thus q :R !R. Show that the derivative is given by T Dq(x) = (A +A )x: 2 3. Let V =C ([1; 1]), with the inner product and induced 2-norm given by Z 1 p hf;gi = f(x)g(x) dx; kfk = hf;fi; 2 1 where f;g2V . 2 (a) Apply the Gram-Schmidt process as presented in class to the set S =f1;x;x ;:::g in order to 1 compute the rst 4 Legendre polynomials , L (x), for n = 0; 1;:::; 4: n (b) The normalization ofkL k = 1. It can be shown thatL (1) =6 0 forn = 0; 1;:::, thus there exist n n 1 constants c =6 0 such thatfP (x)g such that (i)P (x) = c L (x) and (ii)P (1) = 1. Find n n n n n n n=0 P (x) through P (x). 0 4 1 nd (c) It can be shown that these rescaled Legendre polynomials fP (x)g satisfy the 2 order n n=0 dierential equation   d d 2 (1) (1x ) P (x) +n(n + 1)P (x) = 0: n n dx dx Verify this is the case for n = 0; 1 and 4. (d) Note that (1) implies that for  =n(n + 1), the Legendre polynomials satisfy n   d d 2 (2) L[P ] = (1x ) P (x) =n(n + 1)P (x) = P (x); n n n n n dx dx and hence P (x) is the eigenfunctions with corresponding eigenvalue  of the linear operator n n L :V !V given by   d d 2 L[f] = (1x ) f(x) : dx dx 1 Explain why both (1) and (2) also hold forfL (x)g that result from the Gram-Schmidt process n n=0 described above in part (a). 4. Using the normal equations we found the best t least squares line y = a +a x to the three data 0 1 1 points (1; 2), (2; 1) and (3; 3) to bey = 1+ x. Solve this problem alternatively by using the orthogonal 2 projector that results from computing the QR factorization of the original 3 2 coecient matrix. 1 http://en.wikipedia.org/wiki/Legendre_polynomials