{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

m235notes20

# m235notes20 - Notes 20 Eigenvalues Eigenvectors An...

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

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 0 0 λ 2 , λ 1 0 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 ) . It seems reasonable to have: a should be between 0 and 1.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

m235notes20 - Notes 20 Eigenvalues Eigenvectors An...

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

View Full Document
Ask a homework question - tutors are online