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# hw7solutions - Homework 7 Solutions Joshua Hernandez 5.2...

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Unformatted text preview: Homework 7 Solutions Joshua Hernandez November 16, 2009 5.2 - Diagonalizability 2. For each of the following matrices A ∈ M n × n ( R ), test A for diagonalizability, and if A is diagonalizable, find an invertible matrix Q and a diagonal matrix D such that Q-1 AQ = D . b. A = 1 3 3 1 Solution: Computing eigenvalues: χ A ( λ ) = det 1- λ 3 3 1- λ = (1- λ ) 2- 3 2 This polynomial has roots 1 ± 3 = 4 ,-2. Two distinct eigenvalues mean that A is diagonalizable. Then E 4 = N 1- 4 3 3 1- 4 = N-3 3 3-3 = span 1 1 E-2 = N 1- (-2) 3 3 1- (-2) = N 3 3 3 3 = span 1-1 . Our diagonalization is therefore A = 1 1 1 -1 4 0 -2 1 1 1 -1-1 =: QDQ-1 . d. A = 7 -4 0 8 -5 0 6 -6 3 Solution: Computing eigenvalues: χ A ( λ ) = det 7- λ-4 8-5- λ 6-6 3- λ = (7- λ )(-5- λ )(3- λ )- (-4 · 8(3- λ )) = (3- λ )( λ 2- 2 λ- 3) = (3- λ )( λ- 3)( λ + 1) . This polynomial has roots 3 and -1 (a repeated root means that A might not be diagonalizable). E 3 = N 7- 3-4 8-5- 3 6-6 3- 3 = N 4 -4 0 8 -8 0 6 -6 0 = span 1 1 , 1 1 E-1 = N 7- (-1)-4 8-5- (-1) 6-6 3- (-1) = N 8 -4 0 8 -4 0 6 -6 4 = span 2 4 3 . The two eigenspaces have a total dimension of 3, so A is diagonalizable. Our diagonalization is therefore A = 1 0 2 1 0 4 0 1 3 3 0 0 3 0 0 -1 1 0 2 1 0 4 0 1 3 -1 =: QDQ-1 . f. A = 1 1 0 0 1 2 0 0 3 Solution: Since A is an upper-triangular matrix, we can read its eigenvalues off the diagonal: λ = 1 , 3. Then E 1 = N 1- 1 1 1- 1 2 3- 1 = N 0 1 0 0 0 2 0 0 2 = span 1 . We needn’t bother to compute E 3 . The dimension of E 1 is 1, although the root λ = 1 has multiplicity 2. Therefore A is not diagonalizable. 3b. Let V = P 2 ( R ). Define T : V → V by the mapping T ( ax 2 + bx + c ) = cx 2 + bx + a . If diagonalizable, find a basis β for V such that [ T ] β is a diagonal matrix....
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## This note was uploaded on 02/01/2010 for the course MATH math115a taught by Professor Shalom during the Spring '10 term at UCLA.

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hw7solutions - Homework 7 Solutions Joshua Hernandez 5.2...

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