L8_Eigenvectors - MA2213 Lecture 8 Eigenvectors Application...

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Unformatted text preview: MA2213 Lecture 8 Eigenvectors Application of Eigenvectors ufoil 18, lecture 7 : The Fibonacci sequence satisfies ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 n n n n n n s s s s s s L 1 1 5 4 3 2 1 , 5 , 3 , 2 , 1 − + + = = = = = = n n n s s s s s s s s ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − + + + − − + + λ λ λ λ λ λ 1 1 1 1 1 1 1 n n n c c c c − + ± − ± ± ± ± − − + + + = = + = λ λ λ λ λ λ c c c s n n n , , 2 5 1 1 + − − − − + + − − + + ∞ → + ∞ → = + + = λ λ λ λ λ 1 1 1 lim lim n n n n n n n n c c c c s s Fibonacci Ratio Sequence Fibonacci Ratio Sequence Another Biomathematics Application eonardo da Pisa, better known as Fibonacci, invented is famous sequence to compute the reproductive uccess of rabbits* Similar sequences describe frequencies in males, females of a sex-linked gene. or genes (2 alleles) carried in the X chromosome** ,... 2 , 1 , , 1 1 2 1 2 1 2 1 = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + n u u u u n n n n *page i, ** pages 10-12 in The Theory of Evolution and ynamical Systems ,J. Hofbauer and K. Sigmund, 1984. he solution has the form n n v u , n n c c u ) ( 2 1 2 1 − + = here ) ( ), 2 ( 3 2 2 3 1 1 v u c v u c − = + = Eigenvector Problem (pages 333-351) Recall that if v is a square matrix then a nonzero A vector if v Av λ = is an eigenvector corresponding to the λ Eigenvectors and eigenvalues arise in biomathematics where they describe growth and population genetics eigenvalue They arise in physical problems, especially those that involve vibrations in which eigenvalues are related to vibration frequencies They arise in numerical solution of linear equations because they determine convergence properties Example 7.2.1 pages 333-334 For ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 25 . 1 75 . 75 . 25 . 1 A the eigenvalue-eigenvector pairs are ) 2 ( 2 ) 1 ( 1 2 1 v c v c x x x + = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = = 1 1 , 2 ) 1 ( 1 v λ e observe that every (column) vector and ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = = 1 1 , 5 . ) 2 ( 2 v λ where 2 / ) ( 2 1 1 x x c + = 2 / ) ( 1 2 2 x x c − = Example 7.2.1 pages 333-334 Therefore, since x Æ Ax is a linear transformation ) 2 ( 2 ) 1 ( 1 ) 2 ( 2 ) 1 ( 1 ) ( Av c Av c v c v c A Ax + = + = and since ) 2 ( ) 1 ( , v v ) 2 ( 2 2 ) 1 ( 1 1 ) 2 ( 2 ) 1 ( 1 v c v c Av c Av c λ λ + = + are eigenvectors We can repeat this process to obtain ) 2 ( 2 2 2 ) 1 ( 2 1 1 ) 2 ( 2 2 ) 1 ( 1 1 2 ) ( v c v c v c v c A x A λ λ λ λ + = + = ) 2 ( 2 1 2 ) 1 ( 1 ) 2 ( 2 2 ) 1 ( 1 1 ) ( 2 v c v c v c v c x A n n n n n + = + = λ λ uestion What happens as ∞ → n ? Example 7.2.1 pages 333-334 eneral Principle : If a vector v can be expressed as a...
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This note was uploaded on 01/06/2012 for the course MA 2213 taught by Professor Michael during the Fall '07 term at National University of Singapore.

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L8_Eigenvectors - MA2213 Lecture 8 Eigenvectors Application...

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