A general N-dimensional vector consists of N values. They can be arranged as a column or a row and can be real or complex.

Transcription

1 Lnr lgr Vctors gnrl -dmnsonl ctor conssts of lus h cn rrngd s column or row nd cn rl or compl Rcll -dmnsonl ctor cn rprsnt poston, loct, or cclrton Lt & k,, unt ctors long,, & rspctl nd lt k h th componnts ) of th ctor, hn th unt ctors r:,, nd k nd w cn wrt Vctor Spcs st of ctors,,, forms lnr ctor spc f: ) ) ) &, th st s closd undr commutt nd ssoct ddton) ) µ λ µ λ λ λ λ ), ) nd ) ) λµ µ λ, th st s closd undr sclr multplcton nd s oth ssoct nd dstrut) ) h ctor sts such tht 4) Multplcton ls th ctor th sm th orgnl, ) 5) Ech ctor,, hs ngt ctor, such tht ) or W cn wrt thn ) holds

2 w ctors,, tht cn constructd from,,, n s th spn of th st For choc of α, α, α,, αn w cn wrt n α α α α n If for choc of α, α, α,, αn w fnd tht α α α α nn, thn th st,,, n s lnrl dpndnt, ut f no comnton of α, α, α,, αn producs thn th st s lnrl ndpndnt Bss Vctors h ctors,,, form n -dmnsonl ss f th r lnrl ndpndnt hn n othr ctor ddd to th st n th sm -dmnsonl spc) must stsf whr λ, nd w could wrt α k k k λ α k k λ k hs lds to th qult of two ctors Lt k k nd k k k k thn k k ) k f nd onl f k k for k,,,, k Innr Product h nnr product of hs th followng proprts: ) < >< >, nd ) < λ µ c > λ < > µ < c > h ss st,,,,,, s orthonorml f < > δ h, quntt δ s th Kronckr dlt) Hnc, f k k nd k k thn k k < > < > < > δ lso

3 > < > >< < or > < δ ot tht ths dfntons cn tndd to compl ctors Othr proprts of th nnr product nclud: ) h Schwrt nqult: > < hs s somtms wrttn s > < ) rngl nqult: Lnr Oprtors lnr oprtor,, ssocts ctor wth ctor whr so tht µ λ µ λ ) If w us ss ctor must rsult n lnr comnton of th ss ctors or whr s th componnt of, thn or ` If longs to th ss st f, M,,,, thn M f

4 4 Proprts of Lnr Oprtors ) B) B, λ ) λ), B) B) ) O O, I ) I Mtrcs For th componnts of cn rrngd n rctngulr rr M M M s th M mtr hs lmnts If M t s squr mtr of ordr Vctors cn lso rprsntd s mtr ot: r ),,, ) s th trnspos h trnspos of would h th rows nd columns rrsd ) If dffrnt ss s usd th componnts of ctor com ut t s stll th sm ctor Mtr lgr For ctor w cn wrt B) B B) B nd ) λ λ) λ λ

5 B) k B) k k Bk B) k hs r th ruls for mtr ddton, multplcton sclr, nd mtr multplcton, rspctl For mtr ddton 5 S B or S B k k k B k If or S S S B B B S B S S S B B B S B B B B B B For sclr multplcton λ λ λ λ λ λ λ λ Sutrcton D B ) B D B Mtr multplcton s mor dffcult Consdr mtr-ctor multplcton,,,, M M M M M

6 6 For unt ss ctor whr whr th th componnt s on ll th othr componnts r ro hn 4 5 M M M M If w put ll th ctors n mtr n ordr s columns w would h n dntt mtr dfnd low) nd pr-multplng should produc Consdr now th mtr multplcton mpl B B P P B B P P B B whr P B B B B B B B B B P P nd P It cn rdl shown tht: ) Mtr multplcton s ssoct: B C ) B) C B B B ) In gnrl mtr multplcton s not commutt: B B ) Mtr multplcton s dstrut: B) C C B C, nd C B) C C B h null mtr, O, hs ll of ts lmnts qul to ro Hnc, O O O, nd O O whr O hr s lso th dntt mtr, I, whr I I

7 7 h mtr I hs lmnts Rcll ths s th Kronckr dlt, δ, ot tht I must squr nd I Mtr Functons W cn dfn powrs n n tms Of cours must squr ow lor srs cn us to dfn functons h would h th form: whch s n gnrl functon For mpl, S n n n p ) n n! In smlr mnnr othr functons such s sn ) or cos ) Fnll w cn nclud ln I n ) ) n n n rnspos of Mtr h trnspos of wll dnotd s whr th lmnts r rltd

8 8 For mpl, f thn 4 4 ow consdr B) k Bk k Bk Bk k B k k B k k k k hn nd B C ) C B, B C G ) G C B rc of Mtr h trc of mtr,, s th sum of th dgonl componnts, tht s r ) Of cours,, must squr ot: r B) r ) r B), r B) r ) r B) nd r λ ) λr ) r B) B) or Smlrl lso B B B B B r B) r B ) r B C ) r C B ) r C B), ) tht s n cclc prmutton of th mtrcs n th product wll kp th trc th sm

9 9 Dtrmnnt of Mtr h dtrmnnt ppls onl to squr mtrcs nd s dfnd s th sum of ll possl prmuttons of th products of th lmnts of whr th sgn of ch trm s post or ngt ccordng to whthr th prmuttons post or ngt rspctl For mpl for mtr Product Row Ordr Row Prmutton Column Ordr Column Prmutton Fnl Prmutton )) - )-)- )) - )-)- )) - )-)- nd But w cn lso wrt t s hs s n pnson mnors Lplc pnson), nd uss th cofctor mtr nd th mnor

10 h mnor, M, of th lmnt from th mtr s dfnd s th dtrmnnt tht rmns ftr row nd column r rmod h cofctor, C, of lmnt s th mnor mtr, M, multpld ) W cn now wrt th pnson of th dtrmnnt pndng long column s ) k M k k Snc w could lso sum or thn w cn pnd long row lso ) k M k k In ordr to pnd n mnors w nd to contnu from th thr dtrmnnts w hd Epndng mnors Whr, of cours s Gomtrc Intrprtton of Dtrmnnt Consdr th prlllppd t th rght h r of th prlllogrm c, ) s th solut lu of th cross product c csnθ nd th ctor c ponts prpndculr, n rght hnd sns, to th pln of th prlllogrm Proctng th r, wth ts drcton, on th ctor gs th olum s c hs ctl th dtrmnnt, or ) c) c c c

11 Proprts of Dtrmnnts h followng proprts hold: ) hs lso mpls tht whtr holds for th rows holds for th columns whr s n ) Intrchngng two rows or columns) chngs th sgn onl ) If two rows or columns) r th sm th dtrmnnt s ro 4) B B f oth mtrcs r squr Hnc for n numr of squr mtrcs G B G B 5) ddng constnt multpl of on row or column) to nothr ls th dtrmnnt unchngd 6) Multplng row or column) constnt multpls th dtrmnnt th constnt Empl Stp 4 Stps: ) fctorng out from column, ) col col col ) col 4 col 4- col Epndng row ) Epndng row nd thn pndng th dtrmnnts [ ] 6 4) ) ) ) You mght wnt to chck ths! I dd usng Mcsm I thnk I got t

12 Compl Vctors & Mtrcs Lt ) * th compl conugt thn f, whr, * ot tht * ) ) ) ) h nnr product s dfnd < > * Wth rl numrs t would ust th dot product h proprts of th nnr product nclud: ) < >< >, nd ) < λ λc > λ < > λ < c > Hnc, for mpl, < λ µ c > λ* < c > µ * < c > Mtrcs wth compl lus nd ttr dfnton for th trnspos hs s th Hrmtn conugt dont) of, dfnd Hnc *) ) * Empl: G B B G ) hn * nd

13 Homwork o 7 For th squr mtr wth lmnt,,,,, Wht s th lu of th dtrmnnt for ll ordrs > nswr: For Fnd: ) B, ) B, c) 5 4, 6 4 B, d) B, nd ) r B) 5 4 7