c35 - 18.03 Class 35 May 8 The companion matrix and its...

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

View Full Document Right Arrow Icon
The companion matrix and its phase portrait; The matrix exponential: initial value problems. [1] We spent a lot of time studying the second order equation x" + bx' + kx = 0 and if b and k are nonnegative we interpreted them as the damping constant and spring constant (divided by the mass). The companion system is obtained by setting x' = y y' = - kx - by whose matrix of coefficients is the "companion matrix" A = [ 0 1 ; -k -b ] Note that tr(A) = -b , det(A) = k, and the characteristic polynomial of A is lambda^2 + b lambda + k, i.e., it is the same as the characteristic polynomial of the original second order equation. If lambda_1 is an eigenvalue of a companion matrix, then to find an eigenvector we look for v such that [ -lambda_1 , 1 ; * , * ] v = 0 v = [ 1 ; lambda_1 ] does nicely. This makes sense: x = e^{lambda_1 t} has derivative x' = lambda_1 e^{lambda_1 t} , so [ x ; x' ] = e^{lambda t} [ 1 ; lambda_1 ] is a solution to the companion system. QUESTION 1: What region in the (tr,det) plane corresponds to c > 0, k > 0? Ans: the upper left quadrant. QUESTION 2: What region in the (tr,det) plane corresponds to overdamping? Ans: The part of the upper left quadrant which is below the critical parabola, where there are stable nodes. For example x" + (3/2)x' + (1/2)x = 0 is overdamped. We saw long ago that solutions to overdamped equations decay as time increases, can cross the x = 0 slope). axis at most once, and can have at most one critical point (zero
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

c35 - 18.03 Class 35 May 8 The companion matrix and its...

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