Ma1bAnHW6

Ma1bAnHW6 - p x = det xI-A is a polynomial of degree n say p x = ∑ n i =0 a i x i Then p c = ∑ i a i c i On the otherhand cI-A ∈ M n F is a

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CALIFORNIA INSTITUTE OF TECHNOLOGY Department of Mathematics Math 1b, Analytical; Homework Set 6 Due: 10am, Monday, February 25, 2008 Read Chapter 4 in the text and read the notes on eigenvalues and eigenvectors on the course web page. You can collaborate on the problems as long as you write up all solutions in your own words and understand those solutions. Remember to justify all your answers. 1. Let A be an n by n matrix and p ( x ) = char A ( x ) the characteristic polynomial of A , defined as in the class notes rather than as in the text. Prove that for each c F , p ( c ) = det( cI - A ). This shows the polynomial function p : c 7→ p ( c ) is the function c 7→ det( cI - A ), which the text defines to be the characteristic polynomial of A . Note that there is really something to prove here. By definition in the class notes,
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Unformatted text preview: p ( x ) = det( xI-A ) is a polynomial of degree n , say p ( x ) = ∑ n i =0 a i x i . Then p ( c ) = ∑ i a i c i . On the otherhand cI-A ∈ M n ( F ) is a matrix with entries in F , and it is not obvious that the determinant of this matrix is ∑ i a i c i . 2. Problem 6 on page 101 of the text. 3. Problem 6 on page 107 of the text. The matrix A in the problem is to be viewed as the linear map T A : v 7→ Av on the space M 3 , 1 of column vectors. Also determine the eigenvalues and eigenspaces of T A . 4. Problem 4a on page 113. As in the third problem view the matrix A as the linear map T A when calculating eigenvectors. 1...
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This note was uploaded on 04/28/2010 for the course MATH 1B taught by Professor Aschbacher during the Winter '07 term at Caltech.

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