math54 - quiz 11 - solns

# math54 - quiz 11 - solns - Math 54 quiz 11 Solutions 1...

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Math 54 quiz 11 Solutions 1. Consider the matrix A = 2 0 0 2 2 0 0 1 1 , and the associated system of diﬀerential equations ~x 0 = A~x . (a) (3 pts) Find all eigenvalues and eigenvecors of A , and use these to ﬁnd two linearly independent solutions to ~x 0 = A~x . Solution : This is a (lower) triangular matrix, so we can read the eigenvalues oﬀ the diagonal, they are 2 , 2 , 1. For eigenspaces, we have E 1 = Nul 1 0 0 2 1 0 0 1 0 = Nul 1 0 0 0 1 0 0 0 0 = Span 0 0 1 and E 2 = Nul 0 0 0 2 0 0 0 1 - 1 = Nul 1 0 0 0 1 - 1 0 0 0 = Span 0 1 1 These give the solutions 0 0 e t , and 0 e 2 t e 2 t (b) (4 pts) Find a third linearly independent solution to ~x 0 = A~x . Use the funamental matrix this gives you to compute e tA .

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Solution: E 2 is dimension 1, but 2 has multiplicity 2, so we need to ﬁnd a generalized eigenvector with eigenvalue 2. Such a vector would
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## This note was uploaded on 09/17/2011 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at Berkeley.

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math54 - quiz 11 - solns - Math 54 quiz 11 Solutions 1...

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