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Lecture21filled

Lecture21filled - University of Toronto at Scarborough...

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Unformatted text preview: University of Toronto at Scarborough Department of Computer 85 Mathematical Sciences MAT A23 Winter 2008 Lecture 21 Diagonalizabie Matrices Sophie Chrysostomou DEFINITION 0.1 Let A be an n X n matrix with characteristic polynomial p(x\) and distinct eigenvalues A1,/\2, - -- ,Ar. Suppose that (A — Al)” ls a factor of p(/\), but (A —— Agar“ ls not, then we say that cu is the algebraic multiplicity of Ag. If dim(EAi)m ﬁg, then we say that ﬁz- is the geometric multiplicity of Al. THEOREM 0.2 The geometric multiplicity of an eigenvalue of an n ><n matrix A. ls less than or equal to its algebraic multiplicity. 3 1 0 EXAMPLE 0.3 Let A = l 0 3 1 ] Find all the eigenvalues of A. For 0 0 5 each eigenvalue ﬁnd its algebraic and geometric multiplicity. 3a u o til—m: o H 1 =(‘r—M“(S—OO o 0 Sex it???) has {the algebraic wul—blpllol‘w oJE’L A16 has the Olﬂ€bY0Lic Wuluphewﬂ 0F! . I o o 1 o XW" :7 0 l m" o o l X’M‘O Ella-S? O 2 U o o X1733 U dim(h;)=l 15 13M, geometric multiplim o‘F AA 1:: geometric, Mulﬁfliu'rﬂ 34: 3M 4:. algebvaie wwﬁyhoﬂj o'F Jail. Jaw/'55 X3.— 3 I ,2 i o "*4 0 ‘3’ _. .. A’SI: 0 “Z ‘ 8v 0 ,2 l X13377, ELL“SF([7;D o 0 O o 0 0 lew-ms/q. ‘ LP d§WCEW)1[ZBW. wmlt u—Fﬂ; = M3. \Mmk’, VF)» THEOREM 0.4 Let A be an n X rt matrix with a characteristic polynomial MA) = (A W“ A0019 m A2)“2 - - - (A m Adam. Then, A is diagonaiizable m algebraic multiplicity of Ag» = geometric multiplicity of Ai for all i 2 1,2,--- ,m W: A is mm Wmn chow Man P(J\)=(Jx*>u)°“(lwlz)”NCJv-Jm)” awn; H: M3- 054::- OJB- \MUH. 9“: AA: POO has exam n :~ on+oli+~v+otw=n Ly; (’24; \wa 3W- hawk. o-FJW, b3 ﬂaw»- in Previous Page, (94,25 etc. ” JR?“ Lac (3%. be. as. basis {BY by» aqwh 034; has (54} W5 ‘3“ H3 4’ TOJKR, W um‘w 047 Mi 34; A diagonmhzaHe¢1¥A has n lin. {hole/f eigejrl...vewtor$¢>Q® how 11. \in._iv\d_Q/[J “3,9450%. #37 5141514" ~--+(9m=‘n é=> nsc,‘+(‘h+~~ +6.,“ 9011+dv+"'+otmr—n (séid'v' (BY «iﬁi,1,~~;n rm WV, 27X new m3 wk o—FZ #gew. WH: am a 0 S _; A 1g, wot diagonﬁjzabm 3 —3 —3 EXAMPLE 0.5 Let A m —3 3 ——3 :1 . Determine ifA is diagonaiizable. m3 —3 3 If it is, ﬁnd am invertible matria: P, and a diagonal matrix D such that P"1AP = D. 5 m2 0 0 EXAMPLE 0.6 LetAm mg g g 2 . Determine th ts diagonattz— 0 O 4 T" able. [fit is ﬁnd am invertible matrix P, anda diagonalmatttml) such that P EAP: D characteristic, Wlﬂnommt 0“: A 15 P()\}:lAr)~1‘:=‘-(5*J\Qé:9*l 0 (-1) "'1 0 «1 SA 0 0 o '1’)» 4 o ‘1')» o 0 “PA 4 o q. 1% o q. o 0 q 1»). =(5ax) \M L? \+2-(«1) H tr Lt PA = cs—M'L- [mo 151-» ékﬂ‘tvmtsbk—[U' items-m 1+] == (t—w-t m-«M A») (sax-ax Sﬂxﬂw): (Wm u—«Mtwowoo WA U; A) (H ML‘I A) aiggnvomey l\"30'1:0~\3€bmic, WM ‘Dﬂicﬁﬂz l1?“ 0‘? (M52. Wm»: t$35] 0'? edge qu. I U“ smug O'F LA 31) dim(E)\n)7~l=_9€0W‘- WNW.-O‘F;:l\$ 0136 MAR. 3c; )0, have wwaL. o‘F :1. ’ M3 C- geom- 'W‘Ml‘t. 0": .1— =n S i 0 0 ° 1 0 0 ﬁrst ﬂ ° HO 26> t] M- 3 x KHFHHH E»- 6M- a x z” ....- 4 Q- L» o o 0 0 ~ ‘ J? >47. 1, o \ 0 q. 1% Aw? ...
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