A scalar t is an eigenvalue of A if and only if t satisfies the characteristic equation of A: det (A ti) =0

Transcription

1 Chapter 5 a glace: Let e a lear operator whose stadard matrx s wth sze x. he, a ozero vector x s sad to e a egevector of ad f there exsts a scalar sch that (x) x x. he scalar s called a egevale of (or ). scalar t s a egevale of f ad oly f t satsfes the characterstc eqato of : det ( ti) matrx s dagoalzale (.e., 9 vertle, D dagoal that D ) f ad oly f has learly depedet egevectors. sch

2 Chapter 6 revew: (page of ) ) sset S of R s sad to e a orthogoal set f 8, S, 6 ). orthogoal set wthot zero vectors s learly depedet. ) For a sspace W of R, a orthogoal ass ca e fod y startg wth ay ass ad performg Gram-Schmdt rocess. 3) orthoormal ass s a orthogoal ass whose vectors have t orms. 4) Let W e a sspace of R wth a orthoormal ass {w, w,, w k }. he ay vector R ca e qely decomposed as w z where w W ad z W?. orthogoal complemet 5) he orthogoal proecto w ca e fod as w ( w )w ( w )w ( w k )w k

3 Chapter 6 revew: (page of ) 6) other way of fd the orthogoal proecto of R o a sspace W (sg a ass B {,,, k } whch s ot reqred to e orthogoal or orthoormal): w C(C C) C where C ( x k) cotas the ass vectors of W. 7) x matrx Q s called orthogoal f ts colms form a orthoormal ass of R. ( Q Q I ) 8) lear operator o R s called orthogoal f ts stadard matrx s orthogoal. It s also orm-preservg ( (), 8 R ) ad preservg dot prodcts ( v () (v), 8, v R ). 3

4 R R v w v v 3 w w 3 What s a good ass? ) From the pot of vew of a lear operator: ) From the pot of vew of a sspace (or a vector space): orthogoal, or eve orthoormal. 4

5 Secto 6.6 Symmetrc Matrces Defto ( Chapter ) sqare matrx s called a symmetrc matrx f. I ths secto, we wll stdy some terestg propertes of ay symmetrc matrx. I partclar, we wold lke to lear propertes of egevales ad egevectors of a symmetrc matrx. Qestos: () Is a egevale of always real? () re ay two egevectors of correspodg to dstct egevales always orthogoal? (3) Is a symmetrc matrx always dagoalzale? 5

6 Secto 6.6 Symmetrc Matrces Example: Cosder a c R *roposto: If R, the all egevales of are real. roof Let x x, where x [ x x x ]. Sce a a, x x X det( ti ) t ( a c) t ac. Sce ( a c) 4( ac ) ( a c) 4, has two real egevales. X a x x X a x X X a x x x x 6

7 heorem 6.4 roof 7

8 heorem 6.4 roof Let, v R e egevectors of correspodg to egevales, µ, respectvely. v v ( v) v v v µv µ( v). ( - µ) v v sce - µ. heorem 6.4 (*) * roof Follow the proof for heorem 6.4, wth R chaged to C ad chaged to H. 8

9 heorem 6.5 For a matrx R, s symmetrc (.e., ) f ad oly f there s a orthoormal ass for R cosstg of egevectors of, whch case there exsts a orthogoal matrx ad a dagoal matrx D sch that D. 9

10 heorem 6.5 For a matrx R, s symmetrc (.e., ) f ad oly f there s a orthoormal ass for R cosstg of egevectors of, whch case there exsts a orthogoal matrx ad a dagoal matrx D sch that D. roof Sffcecy ( f ): ( ) - D - ( - ) - D D (D ) ( ) D D. Necessty* ( oly f ): By dcto o. he ecessty ovosly holds for. ssme the ecessty holds for, ad cosder R () (). has a egevector R correspodg to a real egevale., ad a orthoormal ass B {,,, } for R y the Exteso heorem ad Gram-Schmdt rocess.

11 [ ]. ad sce, S B B Let B [ ]. B s orthogoal ad S S R a orthogoal C R ad a dagoal L R sch that C SC L y the dcto hypothess. D L SC C C S C C B B C dagoal orthogoal orthogoal

12 heorem 6.5 roof Follow the proof for heorem 6.5, wth R chaged to C ad ( ) chaged to ( ) H. Note B ad C are tary, ad L R. Fdg a orthoormal ass cosstg of egevectors of where R or H C : () Compte all dstct egevales,,, k of. () Determe the correspodg egespaces E, E,, E k. (3) Get a orthoormal ass B for each E. (4) B B B B k s a orthoormal ass for.

13 Example: 5 has egevales 6 ad, wth correspodg egespaces E Spa{[ - ] } ad E Spa{[ ] },_ respectvely. _ B {[ - ] / 5} ad B {[ ] / 5} D, where 6 ad D. 5 3

14 Example: Egevales of are ad 8. E Spa{[ - ], [ - ] } ad E Spa{[ ] }. Ca apply the Gram-Schmdt process to fd orthoormal ases for E ad E. B B D, where 4

15 Example: Coc sectos ad qadratc forms ax xy cy dx ey f. Crcle / ellpse. paraola 3. hyperola How to determe, geeral, the type of coc sectos ased o coeffcets a,, ad c? 5

16 he assocate qadratc form ax xy cy of the qadratc form ax xy cy dx ey f ca e expressed as v v (vʹ ) (vʹ ) (vʹ ) vʹ (vʹ ) Dvʹ (xʹ ) (yʹ ), wth v [ x y ] vʹ [ xʹ yʹ ], a D, ad. c Easy to dge the atre of the correspodg coc secto. 6 Example: x - 4xy 5y - 36, 6 ad. 5 5

17 ± ± ± ± ew ew Note that D Ca always choose a orthogoal wth det, a rotato. 7 x - 4xy 5y - 36 (x ) 6(y ) -36

18 heorem 6.6 (Spectral Decomposto heorem) roof (a) Let [ ] ad D dag[ ]. D [ e e e ] [ e e e ] [ ]. ] [ : the spectral decomposto. 8

19 . ) ( ; ) ( (d). ) ( ; ) ( (c). rak rak () O 9

20 Example: has egevales 5 ad -5. orthoormal _ ass cosstg _ of egevectors of s B {[ - ] / 5, [ ] / 5} {, }.

21 Homework Set for Secto 6.6 Secto 6.6: rolems 5, 8, 9,, 3, 5, 43, 47, 48, 55, 56, 59, 6, 64