FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix


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1 FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if there is a osigular matrix S of the same size such that the matrix S AS is diagoal. That meas all etries of S AS except possibly diagoal etries are zeros. The umbers which show up o the diagoal of S AS are the eigevalues of A. For a diagoal matrix, it is very easy to calculate its powers. Propositio. Let A a diagaalizable matrix of size m m, ad assume that λ λ... 0 S AS λ m for a osigular matrix S. The for a iteger 0 A S λ λ λ m S. Proof. Deote the diagoal matrix by λ λ... 0 Λ, λ m ad observe that, sice it is very easy to multiply diagoal matrices, we have λ λ... 0 ) Λ λ m
2 At the same time sice Λ S AS, we fid that Λ S AS) S AS S AS... S AS S A S because of the obvious cacellatios. We thus coclude that A SΛ S, ad the required idetity follows from ).. Fiboacci umbers ad Kepler s observatio The sequece of Fiboacci umbers F 0,,,, 3,, 8, 3,, 34,, 89, 44,... is defied recursively as follows. Oe begis with F 0 0, ad F. After that every umber is defied to be sum of its two predecessors: F F + F for. The sequece of Fiboacci umbers attract certai iterest for various reasos see, for istace, ). I particular, Johaes Kepler 7 630), oe of the greatest astroomers i the history, observed that the ratio of cosecutive Fiboacci umbers coverges to the golde ratio. Theorem Kepler). F + lim + F I this ote, we make use of liear algebra i order to fid a explicit formula for Fiboacci umbers, ad derive Kepler s observatio from this formula. More specifically, we will prove the followig statemet. Propositio. For F + ) ). We ow show how to derive Kepler s observatio from Propositio. Proof of Kepler s observatio. We simply calculate the limit as follows:
3 3 sice F + lim F as soo as + ) + ) + lim + lim + ) + ) + + ) ) ) ) + ) lim lim + + ) <. + ) ) ) + The rest of the ote is devoted to the proof of Propositio with the help of Liear Algebra, ad Propositio i particular. 3. Liear Algebra iterpretatio of Fiboacci umbers Let L be the liear operator o R represeted by the matrix A 0 with respect to the stadard basis of R. For ay vector v x, y) T, we have that x x + y Lv). 0) y x I particular, for the vector u k whose coordiates are two cosecutive Fiboacci umbers F k, F k ) T, we have that Fk Fk Fk + F Lu k ) A k Fk+ u F k 0 F k F k F k+. k Thus we ca produce a vector whose coordiates are two cosecutive Fiboacci umbers by applyig L may times to the vector u with coordiates F, F 0 ) T, 0): F+ ) A F 0 Equatio is othig but a reformulatio of the defiitio of Fiboacci umbers. This equatio, however, allows us to fid a explicit formula for Fiboacci umbers as soo as we kow how to calculate the powers A of the matrix A with the help of the diagoalizatio.
4 4 4. Diagoalizatio of the matrix A ad proof of Propositio We begi with fidig the eigevalues of A as the roots of its characteristic polyomial λ pλ) deta λi) det λ 0 λ λ. We make use of the quadratic formula to fid the roots as 3) λ + ad λ, ad we coclude, sice the two eigevalues are real ad distict, that the matrix A diogaalizable. I order to diogaalize it, we eed to fid a basis which cosists of eigevectors of the liear operator L. Let us fid the eigevectors from the equatios Lv ) λ v, ad Lv ) λ v or, i coordiates with respect to the stadard basis of R, x x x x λ 0 y, ad λ y 0 y. y We solve these equatios ad fid eigevectors: x λ x, ad y y λ The trasitio matrices betwee the stadard basis ad the basis of eigevectors is thus λ λ S ad S λ λ λ λ λ λ λ λ λ λ λ λ λ λ We ca ow check that, as expected, 4) S AS λ 0. 0 λ We are ow i a positio to prove Propositio with the help of the diogaalizatio 4). Proof of Propositio. Propositio ow implies that A λ S 0 0 λ S, ad we combie this with equatio ) to obtai that F+ A λ S 0 F 0 0 λ S 0 λ λ λ 0 λ λ λ 0 λ λ 0 λ + λ + ) λ λ λ λ.
5 Equatig the etries of the vectors i the last formula we obtai i view of 3) that F λ λ + ) ) λ λ as claimed i Propositio. Remark. Usig the explicit formula from Propositio oe may address some other questios about Fiboacci umbers. Remark. It was Liear Algebra, specifically the diagoalizatio procedure, which allowed us to obtai the explicit formula i Propositio. This is ot the oly way to prove the formula. Remark 3. The sequece of Fiboacci is a very simple example of a sequece give by a recursive relatio. Oe may apply similar methods i order to ivestigate other sequeces give by recurrece relatios.
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