{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Exam_4exam_ _

# Exam_4exam_ _ - MATH 405 Final Exam W Adams Show all work...

This preview shows page 1. Sign up to view the full content.

MATH 405 Final Exam W. Adams December 20, 2002 Show all work. Justify all answers. This is an open book exam. 1. (20 points each) Let A = " 0 i - i 0 # . (a) Show that A is unitary. (b) Find a diagonal matrix D and a unitary matrix U such that A = UDU * . 2. (15 points each) Assume that the minimal polynomial, m A , and characteristic polyno- mial, P A of a matrix A are as given in the three cases below. Describe or write down all possible Jordan canonical (normal) forms for A . (a) P A ( t ) = ( t - 3) 4 ( t - 2) 3 and m A ( t ) = ( t - 3) 3 ( t - 2). (b) P A ( t ) = ( t - 2) 5 and m A ( t ) = ( t - 2) 3 . (c) P A ( t ) = ( t - λ 1 ) e 1 ( t - λ 2 ) e 2 · · · ( t - λ m ) e m and m A ( t ) = ( t - λ 1 ) e 1 - 1 ( t - λ 2 ) e 2 - 1 · · · ( t - λ m ) e m - 1 for integers e 1 2 , . . . , e m 2. 3. (25 points) Find the Jordan canonical form of the matrix A = - 4 0 9 0 1 0 - 4 0 8 . 4. (25 points) Let V be a vector space of dimension 3 and let T be a linear operator on V . Assume that we have a basis v 1 , v 2 , v 3 of V such that the matrix of T with respect to this basis is the matrix A in the last problem. Show that 3 v 1 + 2 v 3 is an eigenvector for T . What is the corresponding eigenvalue?
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online