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Unformatted text preview: CDS 140A — HOMEWORK 3 SCRIBE FILE SHAUN MACBRIDE MAGUIRE Exercise 2. Do all solutions of the system ˙ x = x + y + z ˙ y = y + 2 z ˙ z = 2 z converge to the origin as t → ∞ ? Proof. This system can be written in matrix form d x dt = A x = ⇒ d dt x y z = 1 1 1 1 2 2 x y z . A is upper triangular, and therefore the eigenvalues are 1 , 1 and 2. Each of these eigenvalues has a negative real part, so E S = R 3 and Theorem 1.5 then states that all trajectories approach the origin. Exercise 4. Find the Jordan canonical form, the S + N decomposition and the matrix exponential for the matrix A = 1 0 0 1 1 0 0 1 Proof. First, notice that this matrix is triangular, so we can read the eigenvalues off the diagonal as 1, 1 and 1 . The eigenvectors corresponding to 1 can be found by computing 1 0 0 1 1 0 0 1 v 1 v 2 v 3 = v 1 v 2 v 3 = ⇒ v 1 = v 1 v 2 + v 3 = v 2 v 3 = v 3 Therefore, two linearly independent eigenvectors for the eigenvalue of 1 are (1 , , 0) and (0 , 1 , 0). The0)....
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 Fall '09
 Marsden
 Linear Algebra, Eigenvalue, eigenvector and eigenspace, Singular value decomposition, Orthogonal matrix, SHAUN MACBRIDE MAGUIRE

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