EXAM. Exam 1. Math 5316, Fall December 2, 2012

Transcription

1 EXAM Exam Math 536, Fall 22 December 2, 22 Write all of your answers on separate sheets of paper. You can keep the exam questions. This is a takehome exam, to be worked individually. You can use your notes. You may use Maple as indicated and turn in hardcopy Maple output with your written answers. There is a Maple worksheet on my website with the matrices already entered. The exam is due Saturday, December 8 by 5pm. You must show enough work to justify your answers. Unless otherwise instructed, give exact answers, not approximations (e.g., 2, not.44). This exam has 8 problems. There are 37 points total. Good luck!

2 6 pts. Problem. In each part, find a matrix P so that P AP is in Jordan Canonical Form. I would not undertake this by hand. There is a maple worksheet with the matrices already entered. A. B Problem 2. Find a matrix P so that P AP = J, where J is in Jordan Canonical Form. Find a real matrix S so that S AS = R, where R is in real Jordan form. Again, not a problem for hand computation! See the Maple worksheet

3 Problem 3. Again, not a problem for hand computation. The following matrix A is diagonalizable, so we have P AP = D, where D is diagonal. Find a matrix S so that A t = S AS Problem 4. Again, not a problem for hand computation. You can use the Maple Gram- Schmidt command to check yourself. Read the help page for that command first. A Use the Gram-Schmidt process on the vectors v, v 2, v 3, v 4 to find an orthonormal basis of R 4. Show the Gram-Schmidt process one step at a time. Find an orthogonal matrix Q and an upper triangular matrix R so that QR. v =, v 2 =,, v 3 =, v 4 =. B Use the Gram-Schmidt process on the vectors v, v 2, v 3, v 4 to find an orthonormal basis of C 4. Show the Gram-Schmidt process one step at a time. Find an unitary matrix U and an upper triangular matrix R so that UR. i i i i 2 i + 2 i i v = i, v 2 = + 2i, v 3 =, v 4 = i i 2i 2

4 5 pts. Problem 5. Again, not a problem for hand computation. You can use the Maple Gram- Schmidt command to check yourself. Read the help page for that command first. The following matrix A is symmetric. Find an orthonormal basis for R 5 consisting of eigenvectors of A. Find an orthogonal matrix Q so that Q t AQ is diagonal Problem 6. Let A be an m n matrix over K. Multiplication by A is a linear transformation K n K m and multiplication by A is linear transformation K m K n. We use the standard scalar product on K n and K m. Show that K m = im(a) ker(a ) K n = im(a ) ker(a), and that these are orthogonal direct sums, e.g., the subspaces im(a) and ker(a ) are orthogonal. Hint: Do the first equation first, suppose that v im(a), see if you can get it in ker(a ). 5 pts. Problem 7. Let V be a vector space over K with a K-inner product. Recall that a linear transformation V V is symmetric if T (u), V = u, T (v), u, v V. Recall that an operator P : V V is a projection operator if P 2 = P (which is the same as saying P acts as the identity on im(p )). We know from previous work that in this case V = im(p ) ker(p ). Show that this is an orthogonal direct sum if and only if P is symmetric. A word on Matrix Norms Let V be a vector space over K. Recall that a norm on V is a function V K: v v that has the following properties. 3

5 . v for all v V and v = v =. 2. λv = λ v for all λ K and v V. 3. The triangle inequality holds: for all v, w V, where v + w v + w. In the following problem it is convenient to use the following norm on K d : x = max{ x i i =, 2,..., d}, x x 2 x =. Kd. x d On the vector space K d d of d d matrices, we use the following norm. If [a ij ], then { d } A = max a ij i =, 2,..., d j= In other words, for each row we form the sum of the absolute values of all the entries in the row, and then we take the largest of these row sums. It s easy to see that if A is small, the entries in A must be small. Indeed, we can form the norm of K d d defined by A max = max{ a i,j i, j =, 2,..., d}, where we take the maximum size of an entry. It s easy to check that A max A d A max, so if one of these two norms is small, the other is forced to be small. The advantage of the norm is the following two inequalities Ax A x, x K d, A K d d, AB A B, A, B K d d. 4

6 5 pts. Problem 8. Let A be a d d matrix over C. Let λ, λ 2,..., λ k be the eigenvalues of A. Strictly for the purposes of this problem, we say that the matrix A is stable if λ j < for j =, 2,..., k, i.e., all the eigenvalues have modulus strictly less that one. In this problem, we re working over C. A. Show that if D is a stable d d diagonal matrix, then D n, n. B. If S is a stable d d matrix which is diagonalizable, then S n, n. C. Let A be a stable d d matrix which is not diagonalizable. Show that A n, n. To do this, use the Jordan Decomposition S + N. Since S and N commute, you can compute (S + N) n by the Binomial Theorem. 5