Math 2371 Spring 2009 Practice Problems IV Problem 1: Let A be a linear map on a finite-dimensional complex Euclidean space. Show that if A + A* > 0 then (a) A is invertible and A 1 + ( A 1 )* > 0 . (b) Eigenvalues of A have positive real parts. Problem 2
Math 2371 Spring 2009 Practice Problems III Due as HOMEWORK on Friday, Jan 23 Problem 1: Define the cyclic shift S on C n as S ( x1 , x 2 ,., x n ) = ( x n , x1 ,., x n 1 ) for any vector I x = ( x1 , x 2 ,., x n ) . (a) Show that S is isometry. (b) Find
Math 2371 Spring 2009 Practice Problems II Problem 1: Linear maps A and B on an Euclidean space are called congruent if there is an invertible map P such that B = P * AP . (a) Show that congruence is an equivalence (b) Is there a linear transformation A s
Math 2371 Spring 2009 Practice Problems VII
Problem 1: Let | . | A , | . |B be two norms on X. Verify which of the following are also norms on X. (a) | x |T = | Tx | A with T an invertible map on X (b) | x |C = maxcfw_| x | A , | x |B (c) | x |D = mincfw
Math 2371 Spring 2009 Practice Problems VIII
Problem 1: Let X be a normed finite-dimensional linear space and R be a linear map of X into itself with | R | < 1 . Then S = I R is invertible and S 1 = k =0 R k .
Problem 2: Prove or disprove:
(a) If A is a l
Math 2371 Spring 2009 Practice Problems IX
Problem 1: Prove or disprove: The Euclidean vector norm (when applied to nxn complex matrices)
n A 2 = aij i , j =1
2
1/ 2
is identical to the spectral norm | A | 2 = maxcfw_ : is an eigenvalue of AA* .
n
Prob
Math 2371 Spring 2009 Practice Problems VI Due Friday, March 20 as HOMEWORK Problem 1: (a) Show that the subset K of all polynomials that are positive on (0,1) is a convex subset of all real polynomials. Is K open or closed? (b) Show that the subset M of
Math 2371 Spring 2009 Practice Problems V Problem 1: True or False ? If x and y are non-zero vectors in a finite-dimensional complex Euclidean space, then a necessary and sufficient condition that there exist a positive map A such that Ax = y is that ( x,