hw3_scribe

hw3_scribe - CDS 140A HOMEWORK 3 SCRIBE FILE SHAUN MACBRIDE...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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)....
View Full Document

This note was uploaded on 02/28/2011 for the course MATH 140A taught by Professor Marsden during the Fall '09 term at Caltech.

Page1 / 2

hw3_scribe - CDS 140A HOMEWORK 3 SCRIBE FILE SHAUN MACBRIDE...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online