Lecture23filled - University of Toronto at Scarborough...

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Unformatted text preview: University of Toronto at Scarborough Department of Computer 85 Mathematical Sciences MAT W inter 2008 Lecture 23 Similar Matrices Sophie Chrysostomou DEFINETION 0.1 If A, B are n X n matrices such that B :- P“1AP for some invertible n X 71 matrix P, then we say that A and B are similar mairices. rm down that A , P7 0% diggonahzaHe. LEMMA 0.2 If A and B are similar, than IA] 2 lBl. VLF: A)V7 I :73 Invarin F Ski» F"AP=E7 :7 Maw-‘Amaw-‘y \M. m= “'9‘; l M mam LEMMA 0.3 I f A and B are similar, than A and B have the 35mm eigenmhws. \‘5"3\1\5\P“‘AP’M”‘VI == P"(APu-M)|=IV"‘<A«)\1)P\ Wort/mes awe NUT wit-lawn»?! mwfim = m'mmll vi =\'A~J\1\ [97»in Hivmi A, {5 have the, gym/«L eigmvaluzy E THEOREM 0.4 The Cayley~Hamilton Theorem: Let/1 be an mm matm with, characteristic polynomial 300) m ] AMA] l :2 an)? + ammmml +- - -~1~ (mm-Hm, then p(A)=anA”+anm1A”"i+---+a.1A+a01=0 0 mm matrix. 2 0 3. 1 Howe/work 3 5 4 1 mmwg'g A: “4 m3 —3 w1 1 0 1 2 2"“. M212 ‘1 A: 020' A’AI: O "‘ " :2’N ‘ O ‘l\10 l ‘ *} iz-‘Ao ( 013‘ o 002 "‘ ‘ 2’)‘ gnaw-(’sz >- (1-00” = (100” :— fi'w—gfi—ntyfl» azx—M, ‘53 C—H Thm: A“; 23 WflQA’WvA +1 b1:- 04,” IL 1:: ~ Ans A114 PH‘yz A :W —| leCA*+—.§_A"-~};A‘+1IX .. 'v "7 ~(‘7‘ng"+%A ~ 33A+Z]) A .4 liver?» 4‘0er 200! :0 a 3'71; <7 23 0’17, +«é u 4ov$wéf1°4+1 I “WW—3 “i7. -lf_(+ (1...; I‘lo 0 I 0 0 0 g a 6) (9 Li- 0 0 0L I V1 O V’l/Lé “Tth Emmi) W is diqflonalizab‘c. Nita—JR , 2 ...
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Lecture23filled - University of Toronto at Scarborough...

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