Assignment 7/MATH 247/Winter, 2010 Due: Friday, March 19. Powers of a square matrix

Transcription

1 Assgmet 7/MATH 47/Wter, 00 Due: Frday, March 9 Powers o a square matrx Gve a square matrx A, ts powers A or large, or eve arbtrary, teger expoets ca be calculated by dagoalzg A -- that s possble (!) Namely, A P B P, the A A A A P B P P B P P B P P B B B P P B P ; ad B dag ( λ, λ,, λ ), the (( ),( ) B dag λ λ,,( λ ) ) 0 Example: Let A The char A ( λ ) λ( λ) λ λ ; thus the egevalues o A are λ, (± + 4 (± 5) ; egevectors : P, P ; ad wth P, 5 P, ad ally, A P dag( λ, λ) P P P 0 5 ; (+ 5) 0 A P dag( λ, λ ) P P P 0 ( 5) (+ 5) 0 5 ( ) ( 5) ( + 5) ( 5) (+ 5) ( 5) 5

2 + 5 4( a b ) ( a b ) ( a b ) a b + + where we used the abbrevatos a + 5, b 5 (We also used that (+ 5)( 5) 4 ; ad so (+ 5) ( 5) (+ 5) (+ 5)( 5) (+ 5) ( 4) ; ad smlar urther equaltes) Fboacc umbers ad recurrece equatos The Fboacc umbers,, 3,, are deed as ollows: the rst two terms are deed as,, ad the terms 3, 4, are deed by the recursve ormula + () + + We derve a explct ormula or Thus,, as ollows Let us wrte or the vector,, 3, Let A be the matrx A Next, ote that the equato () gves the vector/matrx equato A + ; () deed, the rst etry the product s 0 + +, whch s the same as +, the rst etry + ; the secod etry the product s + +, whch, by the recurso equato (), s equal to +, the secod etry + Thereore, we have

3 A, A A A A, 3 A A A A, ad geeral (,,3,) A We ca use the explct ormula or A derved above to get: whch gves + 5 a b a b 4( ) ( ) a b a b ( ) Ths ca be rewrtte as ( ( a b ) + a 5 b ) ( a ( + a) b ( +b) ) ; ad 5 [ (+ 5) (3 + 5) ( 5) (3 5) ] 5 Ths s ot a very smple ormula; partcular, t gves the teger terms o rratoal quattes such as 5 But, or large values o, t s a useul way to get, at least approxmately; the orgal recursve ormula requres perormg addtos to arrve at A smlar method wll apply to ay recurrece relato x a x + a x + + a x provded a certa matrx has dstct egevalues; see problem [] below 3

4 [] ) Cosder the matrces 3 A 3 3, B 0 0 Gve ormulas or A ad B or a geeral postve teger ) Cosder the ollowg recurrece relato: x 6x x + 6 (3) x Ths dees a sequece x, x, x3, x4, x5, x 6, whch the rst three terms x, x, x 3 are uspeced, ad every other term s gve by the ormula (3) terms o the three prevous terms Use the method appled above or the Fboacc sequece, ad gve a explct ormula or x, vald or 4, terms o x, x, x 3 (The result wll be smpler tha the oe or the Fboacc sequece) roots 3) Let ( λ) λ a λ a λ a 0 be a polyomal, wth λ, λ, λ Cosder the recurrece relato dstct x a x + a x + + a x (4) Prove that a) or every,,, the sequece x, x λ, x ( λ), s a soluto to the recurrece equato (4) ; ad b) or every sequece { x },, satsyg (4), we have costats c, c, c such that x c λ c λ c ( ) + ( ) + + ( λ ) (5) or all,, (I other words, (5) gves the geeral soluto or (4) ) c) Prove that the characterstc polyomal o the matrx 4

5 A a0 a a a s the gve polyomal ( λ) λ a λ a λ a 0 Ht: c) ca be show drectly, or by rst showg that the vector egevector o A or the egevalue 0 Λ ( λ) ( λ ) λ The asserto s true or ay polyomal λ s a ( λ) λ a λ a λ a, wthout assumg that t has dstct roots d) Use the result b) to derve the explct ormula or the geeral term o the Fboacc sequece dscussed above; ad also or the problem part ) 4) Let A ( ) a j be a upper tragular matrx: a j 0 wheever > j, ad also assume that the dagoal elemets are all the same: a a or all Prove that A s ever dagoalzable uless t s already dagoal: a j 0 wheever j (Ths at least tells us that there are may o-dagoalzable matrces!) [] Gve the geeral real soluto o each o the ollowg systems o deretal equatos: ) A ) B 3) C 5

6 Here, A ad B are the matrces gve [] ); ad C x (O course, ) ad ), s a 3-vector y, o uctos o t : z x xt (), y yt (), z z() t ; ad 3), x y ; x xt (), y yt (), z zt (), w wt ()) z w [3] ) Determe the geeral real soluto o the ollowg system o secod order deretal equatos: x x+ y, y 4x y (6) x ) Fd a soluto y x(0) x(0) y(0) y(0) o the system (6) that satses the tal codtos [4] For all parts ) to 4), A s a (real) symmetrc matrx, wth (ot ecessarly dstct) egevalues λ, λ,, λ, wth correspodg orthogoal system o egevectors P, P, P ) Prove that A ( ), P P λ PP, (O course,, Y deotes the dot-product tr Y ) ) Suppose that each λ 0 Prove that 6

7 , P ( A ( ) P λ PP, ) 3) Assume that λ > or all,, Prove that or every o-zero vector we have that < A 4) Assume that there s such that λ ad there s j such that λ Prove that there s a o-zero vector such that A j 5) For the matrx A, d a o-zero vector A such that 7