Lecture 3: Review of Linear Algebra and MATLAB

Transcription

1 eture 3: Revew of er Aler AAB Vetor mtr otto Vetors tres Vetor spes er trsformtos Eevlues eevetors AAB prmer Itrouto to Ptter Reoto Rro Guterrez-su Wrht Stte Uverst

2 Vetor mtr otto A -mesol (olum) vetor ts trspose re wrtte s: [ ] A (retulr) mtr ts trspose re wrtte s: A he prout of two mtres s A AB where j k k kj Itrouto to Ptter Reoto Rro Guterrez-su Wrht Stte Uverst

3 Vetors he er prout (.k.. ot prout or slr prout) of two vetors s efe, k he mtue of vetor s kk k he orthool projeto of vetor oto vetor s u where vetor u hs ut mtue the sme reto s he le etwee vetors s, os wo vetors re s to e u k / k orthool f orthoorml f A set of vetors,,, re s to e lerl epeet f there ests set of oeffets,,, (t lest oe fferet th zero) suh tht + + Altertvel, set of vetors,,, re s to e lerl epeet f + + k k u θ u Itrouto to Ptter Reoto Rro Guterrez-su Wrht Stte Uverst 3

4 tres he etermt of squre mtr A s A k k+ k Ak ( ) where A k s the mor mtr forme remov the th row the k th olum of A E: the etermt of squre mtr ts trspose s the sme: A A he tre of squre mtr A s the sum of ts ol elemets tr(a) kk k he rk of mtr s the umer of lerl epeet rows (or olums) A squre mtr s s to e o-sulr f ol f ts rk equls the umer of rows (or olums) A o-sulr mtr hs o-zero etermt A squre mtr s s to e orthoorml f AA A AI For squre mtr A f A> for ll, the A s s to e postve-efte (.e., the ovre mtr) f A for ll, the A s s to e postve-semefte he verse of squre mtr A s eote A - s suh tht AA - A - AI he verse A - of mtr A ests f ol f A s o-sulr he pseuo-verse mtr A s tpll use wheever A - oes ot est (euse A s ot squre or A s sulr): A [ A A] A wth A A I ( ssum A A s o - sulr, ote tht AA I eerl) Itrouto to Ptter Reoto Rro Guterrez-su Wrht Stte Uverst 4

5 Vetor spes he -mesol spe whh ll the -mesol vetors rese s lle vetor spe A set of vetors {u, u,... u } s s to form ss for vetor spe f rtrr vetor e represete ler omto of the {u } u + u + u u 3 3 he oeffets {,,... } re lle the ompoets of vetor wth respet to the ss {u } I orer to form ss, t s eessr suffet tht the {u } vetors e lerl epeet j u A ss {u } s s to e orthool f u uj j j A ss {u } s s to e orthoorml f u uj j As emple, the Crtes oorte se s orthoorml se Gve lerl epeet vetors {,,... }, we ostrut orthoorml se {φ, φ,... φ } for the vetor spe spe { } wth the Grm-Shmt rthoormlzto Proeure he ste etwee two pots vetor spe s efe s the mtue of the vetor fferee etwee the pots u 3 u 3 - u E (,) u E(,) k hs s lso lle the Eule ste ( ) k k / u Itrouto to Ptter Reoto Rro Guterrez-su Wrht Stte Uverst 5

6 Itrouto to Ptter Reoto Rro Guterrez-su Wrht Stte Uverst 6 er trsformtos A ler trsformto s mpp from vetor spe X oto vetor spe Y, s represete mtr Gve vetor X, the orrespo vetor o Y s ompute s ote tht the mesolt of the two spes oes ot ee to e the sme. For ptter reoto we tpll hve < (projet oto lower-mesolt spe) A ler trsformto represete squre mtr A s s to e orthoorml whe AA A AI hs mples tht A A - A orthoorml trsformto hs the propert of preserv the mtue of the vetors: A orthoorml mtr e thouht of s rotto of the referee frme he row vetors of orthoorml trsformto form set of orthoorml ss vetors A A (A) (A) j j wth j

7 Eevetors eevlues Gve mtr A, we s tht v s eevetor* f there ests slr λ (the eevlue) suh tht Computto of the eevlues he mtr forme the olum eevetors s lle the mol mtr Propertes Av v ( A I) Av Av A v v s eevetor Y s the orrespo v I ( A I) ( A I) eevlue If A s o-sulr All eevlues re o-zero If A s rel smmetr All eevlues re rel he eevetors ssote wth stt eevlues re orthool If A s postve efte All eevlues wll e postve + v v3 v v v + Chrterst Equto + trvl soluto o trvl soluto Itrouto to Ptter Reoto Rro Guterrez-su Wrht Stte Uverst *he "ee-" of "eevetor" s ormll trslte s "hrterst" 7

8 Iterpretto of eevetors eevlues () If we vew mtr A s ler trsformto, eevetor represets vrt reto the vetor spe Whe trsforme A, pot l o the reto efe v wll rem o tht reto, ts mtue wll e multple the orrespo eevlue λ P P A P v v λ For emple, the trsformto whh rottes 3- vetors out the Z s hs vetor [ ] s ts ol eevetor s the orrespo eevlue z os A s s os v [ ] Itrouto to Ptter Reoto Rro Guterrez-su Wrht Stte Uverst 8

9 Iterpretto of eevetors eevlues () Gve the ovre mtr Σ of Guss struto he eevetors of Σ re the prpl retos of the struto he eevlues re the vres of the orrespo prpl retos he ler trsformto efe the eevetors of Σ les to vetors tht re uorrelte rerless of the form of the struto If the struto hppes to e Guss, the the trsforme vetors wll e sttstll epeet wth v v v fx() ( / / ep (X (X fy() ( ep ) µ v v µ µ µ Itrouto to Ptter Reoto Rro Guterrez-su Wrht Stte Uverst 9

10 AAB prmer he AAB evromet I/ (help ofu) Strt et AAB Cosole I/ Dretor pth he fprtf sprtf omms he strtup.m fle the put omm he help omm Fle I/ he tooloes lo sve omms Bs fetures (help eerl) he fope, flose, fprtf fsf omms Vrles D Grphs (help rph) Spel vrles (,, eps, relm, relm, p, ) he plot omm Arthmet, reltol lo opertos Customz plots Commets pututo (the semolo shorth) e stles, mrkers olors th futos (help elfu) Grs, es lels Arrs mtres ultple plots suplots Arr ostruto Stter-plots ul ostruto he lee zoom omms he : shorth 3D Grphs (help rph3) he lspe omm e plots tr ostruto esh plots ul ostruto me mes omms Cotet rrs mtres 3D stter plots Arr tr e (the olo shorth) the rotte3 omm Arr mtr opertos er Aler (help mtfu) tr elemet--elemet opertos Sets of ler equtos Str rrs mtres (ee, oes zeros) he lest-squres soluto ( A\) Arr mtr sze (sze leth) Eevlue prolems Chrter strs (help strfu) Sttsts Prolt Str eerto Geerto he strmt futo Rom vrles -fles Guss struto: (,) (µ,σ) Srpt fles Uform struto Futo fles Rom vetors Flow otrol orrelte uorrelte vrles f..else..e ostrut Alss for ostrut, m me whle ostrut Vre Covre swth..se ostrut Hstorms Itrouto to Ptter Reoto Rro Guterrez-su Wrht Stte Uverst