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Unformatted text preview: EE263 S. Lall 2009.10.08.01 Homework 3 Due Thursday 10/15. 1. Some basic properties of eigenvalues. 50045 Show that (a) the eigenvalues of A and A T are the same (b) A is invertible if and only if A does not have a zero eigenvalue (c) if the eigenvalues of A are 1 ,..., n and A is invertible, then the eigenvalues of A- 1 are 1 / 1 ,..., 1 / n , (d) the eigenvalues of A and T- 1 AT are the same. Hint: youll need to use the facts that det A = det( A T ), det( AB ) = det A det B , and, if A is invertible, det A- 1 = 1 / det A . 2. Tridiagonal Toeplitz matrices 0660 Some matrices have simple formulae for eigenvalues and eigenvectors. An example we have seen is the circulant matrices. Another example is given by tridiagonal Toeplitz matrices, as follows. Suppose G = b a c b a c b a . . . a c b is an n n real matrix, and a negationslash = 0 and c negationslash = 0. (a) Suppose x = x 1 x 2 . . . x n is an eigenvector of G . Show that cx k- 1 + bx k + ax k +1 = x k for all k = 1 ,...,n , where we define for convenience x = 0 and x n +1 = 0. Hence show that the solution x must be of the form x k = r k 1 + r k 2 for some choice of...
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This note was uploaded on 08/23/2010 for the course EE 263 taught by Professor Boyd,s during the Fall '08 term at Stanford.
- Fall '08