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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 EllaS?
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 lewms/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)”NCJvJm)”
awn; H: M3
054:: OJB \MUH. 9“: AA:
POO has exam n
:~ on+oli+~v+otw=n
Ly; (’24; \wa 3W hawk. oFJW, 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' itemsm 1+]
== (t—wt m«M A») (saxax 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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 Linear Algebra, Algebra

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