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 February 18th, 2018, 10:25 AM #1 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 EigenValues and differential equations.. Please check my math Suppose P is the projection matrix onto the 45◦ line y = x in R2. Its eigenvalues are 1 and 0 with eigenvectors (1, 1) and (1,−1). If dy/dt = −Py (notice minus sign) can you find the limit of y(t) at t = ∞ starting from y(0) = (3, 1)? My Solution: dt/dy = -Py, the eigenvalues are now -1 and 0 and eigenvectors $\displaystyle u_1 = (1, 1)$ and $\displaystyle u_2 = (-1, 1)$ $\displaystyle f(t) = c_1 e^{-t} u_1 + c_2 e^{0t} u_2 \\ f(0) = c_1 u_1 + c_2 u_2 = (3, 1), So. c_1 = 2 ~and~ c_2 = 1 \\ f(t) = 2 e^{-t} u_1 + u_2 \\ f(∞) = u_2 = (-1, 1)$ Am I correct? February 18th, 2018, 11:14 AM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 578 Thanks: 345 Math Focus: Dynamical systems, analytic function theory, numerics You seem to have a good method but you seem to be choosing different eigenvectors in different steps leading to a trivial mistake. If you set $u_2 = (-1,1)$, then your $c_2$ value is wrong. It looks like you computed it with $u_2 = (1,-1)$ instead. It actually doesn't matter which vector you pick but it must be consistent. If you leave your $c_1,c_2$ alone, this corresponds to picking the eigenvector $u_2 = (1,-1)$ which you have initially at the top. In this case, you should get the trajectory limits to $(1,-1)$. You can justify this geometrically as well since $-P$ generates a vector field with all vectors parallel to the vector $(1,1)$. Thus, every trajectory can be computed explicitly by using basic algebra to find the intersection of lines. In this example, the trajectory through $(3,1)$ is a line with slope 1 so it is parameterized by the line $y = x-2$. The trajectory approaches the line $y = -x$ asymptotically with the collision at $t = \infty$ given by the intersection of these lines which is easily seen to occur when $x = 1$ and thus $y = -1$. Thanks from zollen Tags check, differential, eigenvalues, equations, math 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 zollen Calculus 2 October 21st, 2017 09:50 AM rishav.roy10 Differential Equations 0 August 21st, 2013 05:59 AM Norm850 Differential Equations 1 February 29th, 2012 03:26 PM silentwf Differential Equations 5 January 3rd, 2010 03:43 PM knowledgegain Differential Equations 2 April 30th, 2009 09:58 AM

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