College Algebra Exam Review 408

College Algebra Exam Review 408 - A is diagonal. (c) The...

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418 8. MODULES w 2 D .A ± E/w 1 D 2 6 6 4 4 ± 12 ± 6 ± 2 3 7 7 5 . If S denotes the matrix whose columns are v 1 ;v 2 ;w 1 ;w 2 , then S ± 1 AS D J 2 .1/ ˚ J 2 .1/ . Exercises 8.7 8.7.1. Let A D 2 6 6 4 7 4 5 1 ± 15 ± 10 ± 15 ± 3 0 0 5 0 56 52 51 15 3 7 7 5 . The characteristic polyno- mial of A is .x ± 2/.x ± 5/ 3 . Find the Jordan canonical form of A and find an invertible matrix S such that S ± 1 AS is in Jordan form. Use the first method from the text. 8.7.2. Repeat the previous exercise, using the second method from the text. Definition 8.7.18. Say that a matrix A 2 Mat n .K/ is diagonalizable if it is similar to a diagonal matrix. 8.7.3. Show that a matrix A 2 Mat n .K/ is diagonalizable if, and only if, K n has a basis consisting of eigenvectors of A . 8.7.4. Let A 2 Mat n .K/ , and suppose that the characteristic polynomial of A factors into linear factors in KŒxŁ . Show that the following assertions are equivalent: (a) A is diagonalizable. (b) The Jordan canonical form of
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Unformatted text preview: A is diagonal. (c) The minimal polynomial of A has no multiple roots; that is, the minimal polynomial is a product of distinct linear factors. (d) The elementary divisors of A are linear. 8.7.5. Recall that a matrix N is nilpotent if N k D for some k , Let A 2 Mat n .K/ , and suppose that the characteristic polynomial of A factors into linear factors in Kx . Show that A is the sum of two matrices A D A C N , where A is diagonalizable, N is nilpotent, and A N D NA . 8.7.6. Let N 2 Mat n .K/ be a nilpotent matrix. (a) Show that N has characteristic polynomial N .x/ D x n . (b) Show that the Jordan canonical form of N is a direct sum of Jordan blocks J m .0/ ....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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