Linear Algebra

Topic: Linear Recurrences 401Topic: Linear RecurrencesIn 1202 Leonardo of Pisa, also known as Fibonacci, posed this problem.A certain man put a pair of rabbits in a place surrounded on allsides by a wall. How many pairs of rabbits can be produced fromthat pair in a year if it is supposed that every month each pair begetsa new pair which from the second month on becomes productive?This moves past an elementary exponential growth model for population increaseto include the fact that there is an initial period where newborns are notfertile. However, it retains other simplyfing assumptions, such as that there isno gestation period and no mortality.The number of newborn pairs that will appear in the upcoming month issimply the number of pairs that were alive last month, since those will all befertile, having been alive for two months. The number of pairs alive next monthis the sum of the number alive last month and the number of newborns.f(n + 1) = f(n) + f(n − 1) where f(0) = 1, f(1) = 1The is an example of a recurrence relation (it is called that because the valuesof f are calculated by looking at other, prior, values of f). From it, we caneasily answer Fibonacci’s twelve-month question.month 0 1 2 3 4 5 6 7 8 9 10 11 12pairs 1 1 2 3 5 8 13 21 34 55 89 144 233The sequence of numbers defined by the above equation (of which the first feware listed) is the Fibonacci sequence. The material of this chapter can be usedto give a formula with which we can can calculate f(n + 1) without having tofirst find f(n), f(n − 1), etc.For that, observe that the recurrence is a linear relationship and so we cangive a suitable matrix formulation of it.( ) ( )1 1 f(n)=1 0 f(n − 1)(f(n + 1)f(n))where( ) (f(1) 1=f(0) 1)Then, where we write T for the matrix and ⃗v n for the vector with componentsf(n+1) and f(n), we have that ⃗v n = T n ⃗v 0 . The advantage of this matrix formulationis that by diagonalizing T we get a fast way to compute its powers: whereT = P DP −1 we have T n = P D n P −1 , and the n-th power of the diagonalmatrix D is the diagonal matrix whose entries that are the n-th powers of theentries of D.The characteristic equation of T is λ 2 − λ − 1. The quadratic formula givesits roots as (1 + √ 5)/2 and (1 − √ 5)/2. Diagonalizing gives this.( ) ( √1 11+ 5= 21 01− √ 521 1) ( 1+ √ 52001− √ 52) ( 1√5− 1−√ 52 √ 5−1 √51+ √ 52 √ 5)