Math33A_Laub_Spring2012_Final.pdf

Consider the continuous time initial value problem x

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2. Consider the continuous-time initial-value problem ˙ x ( t ) = dx ( t ) dt = Ax ( t ); x (0) = x 0 for t 0. Let A = ° 3 4 ° 1 2 ° 1 4 ° 1 2 and x 0 = 1 1 . (a) Without computing the eigenvalues explicitly, is A asymptotically stable? Why or why not? (b) Determine an explicit expression for x ( t ), i.e., solve the initial value problem. (c) Determine lim t ! + 1 x ( t ). (d) Consider instead the discrete-time problem x k +1 = Ax k ; x 0 = 1 1 for k 0 and with the same matrix A as above. Use your eigenvalue/eigenvector calculations in the continuous-time problem above to solve this initial value prob- lem for x k ; k 0. (e) Determine lim k ! + 1 x k . 2

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Extra space for problem 1 and/or 2. 3
3. (a) For what values of the scalar a , if any, is the matrix A = 2 4 a ° 1 ° 1 ° 1 a ° 1 ° 1 ° 1 a 3 5 pos- itive definite? Show your reasoning carefully, i.e., state explicitly which criterion for positive definiteness you are using. (b) For what values of the scalar b , if any, is the matrix B = 2 4 b 4 0 4 b 3 0 3 b 3 5 positive definite? Show your reasoning carefully, i.e., state explicitly which criterion for positive definiteness you are using. 4

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4. The spectral factorization or spectral representation of a symmetric matrix A 2 IR n £ n is given by A = 1 x 1 x + 1 + · · · + n x n x + n = 1 x 1 x
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