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Unformatted text preview: Notes 20: Eigenvalues, Eigenvectors, An Ecological Model Lecture December 1, 2010 Given a 2 × 2 matrix M , our goal is to find a basis B of R 2 so that the matrix representing M with respect to the basis B is simple. Indeed we will show how to find a basis so that M B is takes on one of the three forms λ 1 λ 2 , λ 1 λ , a- b b a . Here λ 1 ,λ 2 ,λ,a,b ∈ R . We look at the last of these matrices more closely. Let ρ = √ a 2 + b 2 , then we can write a b b a = ρ a/ρ- b/ρ b/ρ a/ρ . Note that ( a/ρ ) 2 +( b/ρ ) 2 = 1 . This means we can find θ so that a/ρ = cos( θ ) ,b/ρ = sin( θ ). Thus the matrix a- b b a = ρ cos( θ )- sin( θ ) sin( θ ) cos( θ ) is the composition of a dilation by ρ and a rotation by angle θ. Coyotes and Roadrunners We begin our analysis of 2 × 2 matrices by examining a model of the interactions of an imaginary coyote population and a road runner population. Let x ( n ) ,y ( n ) denote the populations of coyotes and roadrunners after n periods of time. We assume there is a matrix A = a b c d so that x ( n + 1) y ( n + 1) = a b c d x ( n ) y ( n ) . Lets choose the coefficients a,b,c,d so that this behaves somewhat reasonably. We have the equations x ( n + 1) = ax ( n ) + by ( n ) y ( n + 1) = cx ( n ) + dy ( n ) ....
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