1 Last time: similar and diagonalizable matrices

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1 Last time: similar ad diagoalizable matrices Let be a positive iteger Suppose A is a matrix, v R, ad λ R Recall that v a eigevector for A with eigevalue λ if v ad Av λv, or equivaletly if v is a ozero elemet of NulA λi) The umber λ is a eigevalue of A if there exists some eigevector with this eigevalue If the ullspace NulA λi) is ozero, the it is called the λ-eigespace of A The eigevalues of A are the solutios to the polyomial equatio deta xi) Importat fact Ay set of eigevectors of A with all distict eigevalues is liearly idepedet Two matrices A ad B are similar if there is a ivertible matrix P such that A P BP Similar matrices have the same eigevalues but usually differet eigevectors Example The matrix A is similar to A A matrix is diagoal if all of its ozero etries appear i diagoal positios, ),, ),, or, ) A matrix A is diagoalizable if it is similar to a diagoal matrix I other words, A is diagoalizable if we ca write A P DP where λ λ D λ is a diagoal matrix I this case λ, λ,, λ are the eigevalues of A are if P v v v the Av i λ i v i for each i,,,, ie, the colums of P are a basis for R of eigevectors of A We proved these results last time: Theorem A matrix A is diagoalizable if ad oly if R has a basis whose elemets are all eigevectors of A Theorem If A is a matrix with distict eigevalues the A is diagoalizable Diagoalizatio ad Fiboacci umbers Kowig how to diagoalize matrices will let us prove a exact formula for the Fiboacci umbers The sequece f of Fiboacci umbers starts as f, f, f, f 3, f 3, f, f 6 8, f 7 3 For, the sequece is defied by f f + f We have f ad f The sequece grows expoetially Write log for the atural logarithm The

2 log f 73 log f 736 log f 8 log f 83 log f 8 These umbers seem to be covergig to somethig I fact, if we set x 8 ad e 78 so that log e ) the e x ) Ca we explai this? Defie a f ad f + for If > the Similarly, we compute that if > the a f f + f a + f + f + f + a a + We ca put these two equatios together ito oe matrix equatio: a a Sice this holds for all >, we have a a a 3 a 3 3 a b I other words, a Thus if we could get a exact formula for the matrix the we could derive a formula for a f ad f +, which would determie f for all The best way we kow to compute A for large values of is to diagoalize A, that is, to fid a ivertible matrix P ad a diagoal matrix D such that A P DP, sice the A P D P Note, however, that at the outset it s ot clear if this is eve possible Defie the matrix A To determie if A is diagoalizable, our first step is to compute its eigevalues, which are the roots of the polyomial x deta xi) det x) x) x 3x + x By the quadratic formula, the values of x where this polyomial is zero are α 3 + ad β 3

3 These umbers are the eigevalues of A Sice α β, these eigevalues are distict so A is diagoalizable Note that αβ 3 )3 + )/ 9 )/ Our ext step is to fid bases for the α- ad β-eigespaces of A To fid a eigevector for A with eigevalue α, we row reduce α α α A αi α α α) α) The secod equality holds sice α) α) ) )/ + )/ This computatio shows that x NulA αi) if ad oly if x is a eigevector for A with Av αv v α x α RREFA αi) where x x + α)x, so To fid a eigevector for A with eigevalue β, we similarly row reduce β β β A βi β β β) β) β RREFA βi) The secod equality holds sice also β) β) By algebra idetical to the previous case, we deduce that β w is a eigevector for A with Av βv This meas that for P v w α β ad D α β we have A P DP Sice P is with det P α ) β ) α β, we have D α β ad P β α We therefore have a A P D P α β α β β α Before computig aythig further, it helps to make a few simplificatios Note that α + β ad β α 3

4 Hece a β α α α β β β α α α β β β α α )α β )β α )β ) β α ) α )α β )β Sice α )β ) )+ ), rewritig this matrix equatio gives i) f a α β ) ii) f + α )α β )β ) We ow make oe more uexpected observatio: α ) + ) α ad Thus i) ad ii) become ad β ) ) + 3 β f α ) β ) ) *) f + α ) + β ) +) **) Puttig *) ad **) together gives a commo formula for f for all Sice we get: α + ad β Theorem For all itegers it holds that f + ) Check that this holds eve whe ad ) ) ) ) ) How does this explai our origial umeric observatios? Well, sice 68 has absolute value less tha, it follows that if is very large the f + ) so log f log + ) ) ) log + log + )

5 Therefore Ad ideed Moreover, if x log ) lim log f lim log + log }{{}}{{} log + ) 8 + ) log log f + ) the e x + )/ so e x ) + ) 3 Diagoalizig matrices whose eigevalues are ot distict If a matrix A has distict eigevalues with correspodig eigevectors v, v,, v, the the matrix P v v v is automatically ivertible sice its colums are liearly idepedet, ad the matrix D P AP is diagoal such that A P DP Whe A is diagoalizable but has fewer tha distict eigevalues, we ca still build up P i such a way that P is automatically ivertible ad P AP is automatically diagoal Recall that if λ is a eigevalue of A the NulA λi) is the λ-eigespace of A The multiplicity of the eigevalue λ is the largest iteger m such that we ca write the characteristic polyomial of A as the product deta xi) λ x) m px) for some polyomial px) For example, if A the x deta xi) det x) x) + x x x + x ) x so is a eigevalue of A with multiplicity Theorem Let A be a matrix Suppose A has distict eigevalues λ, λ,, λ p where p The followig properties the hold: a) For each i,,, p, the dimesio of the λ i -eigespace of A is at most the multiplicity of λ i b) A is diagoalizable if ad oly if the sum of the dimesios of the eigespaces of A is c) Suppose A is diagoalizable ad B i is a basis for the λ i -eigespace The the uio B B B p is a basis for R cosistig of eigevectors of A If the elemets of this uio are the vectors v, v,, v the the matrix P v v v is ivertible ad P AP is diagoal Proof Fix a idex i {,,, p} Let λ λ i ad suppose λ has multiplicity m ad NulA λi) has dimesio d Let v, v,, v d be a basis for NulA λi) Oe of the corollaries we saw for the dimesio theorem is that it is always possible to choose vectors v d+, v d+,, v R such that v, v,, v d, v d+, v d+,, v is a basis for R Defie Q v v v The colums of this matrix are liearly idepedet, so Q is ivertible with Qe j v j ad Q v j e j for all j,,, Defie B Q AQ If j {,,, d} the the jth colum of B is Be j Q AQe j Q Av j λq v j λe j

6 This meas that the first d colums of B are λ λ λ so B has the block-triagular form λ λ B λ λid Y Z where Y is a arbitrary d d) matrix ad Z is a arbitrary d) d) matrix Now, we wat to deduce that detb xi) λ x) d detz xi) Sice deta xi) detb xi) as A ad B are similar, ad sice detz xi) is a polyomial i x, we see that deta xi) ca be writte as λ x) d px) for some polyomial px) Sice m is maximal such that deta xi) λ x) m px), it must hold that d m This proves part a) To prove parts b) ad c), suppose vi, v i,, vli i is a basis for the λ i -eigespace of A for each i,,, p Let B i {vi, v i,, vli i } We claim that B B B p is a liearly idepedet set To prove this, suppose p li i j cj i vj i for some coefficiets cj i R It suffices to show that every c j i Let w i l i j cj i vj i R We the have w + w + + w p Each w i is either zero or a eigevector of A with eigevalue λ i Why?) Sice eigevectors of A with distict eigevalues are liearly idepedet, we must have w w w p But sice each set B i is liearly idepedet, this implies that c j i We coclude that B B B p is a liearly idepedet set for all i, j If the sum of the dimesios of the eigespaces of A is the B B B p is a set of liearly idepedet eigevectors of A, so A is diagoalizable If A is diagoalizable the A has liearly idepedet eigevectors Amog these vectors, the umber that ca belog to ay particular eigespace of A is ecessarily the dimesio of that eigespace, so it follows that sum of the dimesios of the eigespaces of A at least This sum caot be more tha sice the sum is the size of the liearly idepedet set B B B p R This proves part b) To prove part c), ote that if A is diagoalizable the B B B p is a set of liearly idepedet vectors i R, so is a basis for R 6

7 Example Cosider the lower-triagular matrix A 3 3 Its characteristic polyomial is deta xi) x) x 3) The eigevalues of A are therefore ad 3, each with multiplicity Sice A I 8 8 it follows that x NulA I) if ad oly if x x x x 3 x so Sice 8 A 3)I A + 3I, 8x 3 6x x 3 + x x 3 x it follows that x NulA + 3I) if ad oly if x x x x 3 x so, x 3 x 8 6 x x RREFA I) is a basis for NulA I) x 3 + x is a basis for NulA + 3I) 6 RREFA + 3I) Each eigespace has dimesio, so the sum of the dimesios of the eigespaces of A is + Thus A is diagoalizable I particular, if P 8 6 7

8 the P is ivertible ad P AP 3 3 ad A P 3 3 P 8