math294_hw9fa06

math294_hw9fa06 - Math 294 - HW9 Solutions 7.1 Discrete...

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Unformatted text preview: Math 294 - HW9 Solutions 7.1 Discrete Dynamical Systems 5. Assume A v = λ v and B v = β v for some eigenvalues λ, β . The ( A + B ) v = A v + B v = λ v + β v = ( λ + β ) v so v is an eigenvector of A + B with eigenvalue λ + β . 6. Yes. If A v = λ v and B v = μ v , then AB v = A ( μ v ) = μ ( A v ) = μλ v . 30. 32. 38. 4 1 1 − 5 − 3 − 1 − 1 2 1 − 1 − 1 = 2 − 2 − 2 = 2 1 − 1 − 1 . The associated eigenvalue is 2. 42. Since A 1 is simply the first column of A , the first column must be a multiple of e 1 . Similarly, the third column must be a multiple of e 3 . There are no other restrictions on the form of A , meaning it can be any matrix of the form a b c d e = a 1 + b 1 + c 1 + d 1 + e 1 . Thus, a basis of V is 1 , 1 , 1 , 1 , 1 . 50. Let v ( t ) = bracketleftbigg h ( t ) f ( t ) bracketrightbigg , and A v ( t ) = v ( t + 1), where A = bracketleftbigg 4 − 2 1 1 bracketrightbigg . Now we will proceed as in the example worked on pp 292-295....
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This note was uploaded on 02/07/2008 for the course MATH 2940 taught by Professor Hui during the Fall '05 term at Cornell.

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math294_hw9fa06 - Math 294 - HW9 Solutions 7.1 Discrete...

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