LAHW#13 Due December 20, 2010

6.1 Eigenvalues and Eigenvectors 5. –The matrix represents a rotations of 90° counterclockwise. Obviously, Ax is never a simple multiple of x (if x ≠ 0), because the vector x and Ax point in different directions. However, the equation Ax = λ x must have nontrivial solutions because the characteristic polynomial must have roots. Explain this apparent contradiction. Also, find the diagonalization of A and check you work.

6.1 Eigenvalues and Eigenvectors 10. –Criticize this argument: With a sequence of row operations, we establish that Therefore, the eigenvalues of A are 1, 3, -122.

6.1 Eigenvalues and Eigenvectors 32. –Argue that the eigenvalues of a matrix A are also eigenvalues of A T. Do A and A T have the same characteristic polynomial? (Explain why or given a counterexample.)

6.1 Eigenvalues and Eigenvectors 35. –Consider a dynamical system where Explain why the sum of the two components of x (r) is a constant, that is, is independent of r (Here each x (r) is a vector in R 2.)

6.1 Eigenvalues and Eigenvectors 37. –Explain why the hypothesis {u 1, u 2, …, u n } is a linearly independent set in R n leads to the conclusion that there is an n × n invertible matrix A having the vectors u i as eigenvectors. Can the eigenvalues also be assigned freely?

6.1 Eigenvalues and Eigenvectors 43. –Let λ be an eigenvalue of an n × n matrix A, and define V = {x | Ax = λx}. This set is called an eigenspace of A. Establish in detail that V is a subspace of R n. Is each vector in V an eigenvector of A ?

6.1 Eigenvalues and Eigenvectors 70. –Let Find all the eigenvalues. Find all the eigenvectors. What are the algebraic and geometric multiplicities? Find the matrices such that AP = PD, where D is a diagonal matrix. Determine the matrices in the diagonalization factorization A = PDP -1. Give a general formula for A k.

6.1 Eigenvalues and Eigenvectors 74. –Show that the Fibonacci sequence x 0 = 0, x 1 = 1, and x k+1 = x k + x k-1 for k ≧ 1 can be written as Determine A k.