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 January 31st, 2015, 06:44 PM #1 Newbie   Joined: Jan 2015 From: Mtl Posts: 1 Thanks: 0 Eigenvalues of piecewise linear systems Hi all, I am a theoretical ecology M.Sc student and I'm struggling with the calculation of the eigenvalues of a fairly simple system. I have a difference equations system where the following 3x3 state matrix: line 1:[0, 1-x, 1-x] line 2:[k1, 0, 0] line 3:[0, k2, k3] is valid over x element of [0,k4] and line 1:[0, 0, 0] line 2:[k1, 0, 0] line 3:[0, k2, k3] is valid over x element of ]k4,inf Would there be a simple way to do a hand calculation of the eigenvalues and eigenvectors of such a system? Could someone point me towards documentation that would indicate how to do such a calculation? Thanks in advance! Julien February 4th, 2015, 05:01 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 Yes, it is reasonably easy to find the eigenvalues "by hand". To find eigenvalues for the first you need to solve the equation $\displaystyle \left|\begin{array}{ccc}-\lambda & 1- x & 1- x \\ k_1 & -\lambda & 0 \\ 0 & k_2 & k_3- \lambda \end{array}\right|= 0$. Expanding on the middle row, that is $\displaystyle -k_1\left|\begin{array}{cc}1- x & 1- x \\ k_2 & k_2- \lambda\end{array}\right|+ \lambda\left|\begin{array}{cc}-\lambda & 1- x \\ 0 & k_3- \lambda \end{array}\right|$$\displaystyle = -k_1((1- x)(k_2- \lambda)- k_2(1- x)+ \lambda(-\lambda(k_3- \lambda)= 0$ a cubic equation for $\displaystyle \lambda$. The second is far easier. Since it is a triangular matrix, its eigenvalues are just the numbers on the "main diagonal", 0 and $\displaystyle k_3$ with 0 as a double eigenvalue. Tags eigenvalues, linear, piecewise, piecewise linear systems, pwl, stability analysis, systems Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Predator Linear Algebra 2 December 4th, 2013 05:11 PM henrymerrild Linear Algebra 2 November 29th, 2013 10:01 AM ricsi046 Algebra 10 September 30th, 2013 12:41 AM tools Linear Algebra 1 September 21st, 2012 12:38 PM cka Calculus 3 August 23rd, 2010 02:10 AM

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