Vectors and Matrices

Transcription

1 Collg of Egrg d Compur Scc Mchcl Egrg Dprm Egrg Alyss Nos Ls updd: Augus 8, 7 Lrry Cro Vcors d Mrcs Iroduco Ths os provd roduco o h us of vcors d mrcs grg lyss. I ddo, hy provd dscusso of how h smpl cocp of vcor mchcs lds o h cocp of vcor spcs for grg lyss. Mr oo s usd o smplfy h rprso of lr lgrc quos. I ddo, h mr rprso of sysms of quos provds mpor proprs rgrdg h sysm of quos. Th dscusso hr prss my rsuls whou proof. You c rfr o grl dvcd grg mh, lk h o y Kryszg or o lr lgr for such proofs. Prs of hs os hv prprd for us vry of courss o provd ckgroud formo o h us of mrcs grg prolms. Cosquly, som of h mrl my o usd hs cours d dffr scos from hs os my ssgd dffr ms h cours. Vcors, Lr Idpdc d Bss Ss A vcor s commo cocp grg mchcs h mos suds frs sw hr hgh-school physcs courss. Vcors r usully dscrd roducory courss s quy h hs mgud d drco. Forc d vlocy r commo mpls of vcors usd sc mchcs cours. I ddo o rprsg vcor rms of s mgud d drco, w c lso rprs y vcor rms of s compos. Ths s llusrd h fgur h rgh. Hr w hv forc vcor, f, wh mgud, f, d drco,, rlv o h f s. No h h oo of h vcor, f, d s mgud, f, r dffr. Th vcor s h full fy spcfco of mgud d drco;.g., pouds forc gl of o from h s. Th f mgud f s pouds hs mpl. Th compos of h vcor h d y drcos r clld f d fy, rspcvly. Ths r o vcors, u r sclrs h r mulpld y h u vcors h d y drco o gv h vcor forcs h coord drcos. Th u vcors h d y drco r usully gv h symols d, rspcvly. I hs cs w would wr h vcor rms of s compos s f = f + fy. Th vcor compos r clld sclrs o dsgush hm from vcors. Formlly sclr s dfd s quy whch s vr udr coord rsformo. Th cocp of wrg vcor rms of s compos s mpor o grg lyss. Isd of wrg f = f + fy, w c wr f = [f fy], wh h udrsdg h h frs umr s h compo of h vcor d h scod umr s h y compo of h vcor. Usg hs oo w c wr h u vcors h d y drcos s = [ ] d = [ ]. Ths oo for u vcors provds lk w rprsg vcor s row or colum mr, s w wll do low, d h covol vcor oo: f = f + fy d f = [f fy]. If w susu = [ ] d = [ ] h quo f = f + fy, w g h rsul h f = f[, ] + fy[, ] = [f fy]. I plc of h oo f d fy for h d y compos, w c us umrcl suscrps for h coord drcos d compos. I hs schm w would cll h d y coord drcos h d drcos d h vcor compos

2 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg would lld s f d f. Th umrcl oo llows grlzo o sysms wh rrry umr of dmsos. From h dgrm of h vcor, f, d s compos, w s h h mgud of h vcor, f, s gv y Pyhgors s horm: f f f y f f. W kow h w c d h wodmsol vcor show o h prvous pg o hr dmsos. I hs cs our vcors hv hr compos, o ch coord drco. W c wr h u vcors h hr coord drcos s = [ ], = [ ], d k = [ ]. W would h wr our hr-dmsol vcor, usg umrcl suscrps plc of, y, d z suscrps, s f = f + f + fk or f = [f f f]. If w susu = [ ], = [ ], d k = [,, ] h quo f = f + f + fk, w g h rsul h f = f[ ] + f[ ] + f[ ] = [f, f, f]. Th do produc of wo vcors, d s wr s. Th do produc s sclr d s vlu s cos, whr s h gl w h wo vcors. Th mgud of h u vcors,,, d k, s o. Ech u vcor s prlll o slf so f w vlu,, or k k, w g cos = for h do produc. Ay wo dffr u vcors r prpdculr o ch ohr so h gl w hm s 9 o ; hus h do produc of y wo dffr u vcors s cos9 o =. Th do produc of wo vcors, prssd rms of hr compos c wr s follows. = + +k + +k = + + k k + k + k + k k = + +. Ths rsul h do produc of wo vcors s h sum of h producs of h dvdul compos s h ss for h grlzo of h do produc o h r produc s dscussd low. Th do produc rprss h mgud of h frs compo log h drco of h scod compo ms h mgud of h scod compo. Th mos fmlr pplco of h do produc s grg mchcs s h dfo of work s dw = f d; hs gvs h produc of h mgud of h forc compo h drco of h dsplcm ms h mgud of h dsplcm. Th fc h h u vcors r prpdculr o ch ohr gvs prculrly smpl rloshp for h do produc. Ths s mpor ool lr pplco of vcors. W us h word orhogol o df s of vcors h r muully prpdculr. I ddo, wh w hv s of muully prpdculr vcors, ch of whch hs mgud of o, w cll hs s of vcors orhoorml s. W c rprs y hr-dmsol vcor rms of h hr u vcors,,, d k. Bcus of hs w sy h hs hr vcors r ss s for rprsg y hr rl, hr-dmsol vcor. I fc, w could us y hr vcors plc of,, d k, o rprs y hr-dmsol vcor so log s h s of hr vcors s lrly dpd. For mpl, w could us w s, m = + + k, = + k d o = + k. Ths would cov s o us, sc h u vcors r o orhogol d h do producs would hrd o compu. Nvrhlss, w could rprs y vcor, = m + + o, sd of h quvl vcor k. W c covr h vcor B = + + k compos o h m,,o ss y solvg h followg s of quos: [] You c vrfy h h grl soluo o hs s of quos s h o show low.

3 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg [] Th wo ss of quos ov llow us o covr w h wo dffr rprsos. Howvr, cosdr h followg s of vcors, m = + + k, = + k d o = +, whr w hv md oly slgh chg o from s prvous dfo s + k. I hs cs w s h vcor, A = m + + o, s qul o k. Wh c ry o covr h vcor B = + + k compos o h m,,o ss y solvg h followg s of quos: [] Howvr, w fd h surcg h frs wo quos gvs h rsul h =, sd of quo h w c solv for or. Thus, w coclud h h s of quos hs o soluo d w co us h proposd s of vcors o rprs y hr-dmsol vcor. Th rso for hs s h h w proposd s dos o hv vcors h r lrly dpd. Isd, h hr proposd vcors ssfy h followg lr quo: m + + o =. Th s, w c solv for o of hs vcors rms of h ohr wo. W wll lr s h y s of vcors h w w o us o rprs y ohr vcor h spc such s of vcors s clld ss s mus lrly dpd. W wll d hs sc cocps of vcors, prculrly h rsoluo of vcor o s of compos, h us of lrly dpd ss s o rprs y vcor h prculr lyss of rs, d h do produc of wo vcors. Ths ds wll lr usd o df grlzd vcor spc h ppls o ss of umrs or fucos whos hvor s smlr o h fmlr physcl vcors from grg mchcs. Frs, w wll dvlop h grl oo of mrcs, whch cluds rprso of vcors rms of hr compos. Mrcs d hr Opros A mr plurl mrcs s rprsd s wo-dmsol rry of lms,, whr s h row d d s h colum d. Th r mr s rprsd y h sgl symol A. I grl, w spk of mr s hvg rows d m colums. Such mr s clld y m or m mr. Equo [] shows h rprso of ypcl m mr. I grl, h umr of rows my dffr from h umr of colums. Somms h mr s wr s A m o show s sz. Sz s dfd s h umr of rows d h umr of colums. A mr h hs h umr of rows qul o h umr of colums s clld squr mr. Mrcs r usd o rprs physcl qus h hv mor h o umr. Ths r usully usd for grg sysms such s srucurs or works whch w rprs collco of umrs, such s h dvdul sffss of h mmrs of srucur, s sgl symol kow s sffss mr. Nworks of pps, crcus, rffc srs, d h lk my rprsd y cocvy mr whch dcs whch pr of ods h mr r drcly od o ch ohr. Th us of mr oo d formul for mrcs lds o mpor lycl rsuls. W wll s h mr propry kows s s gvlus rprss h fudml vro frqucs mchcl sysm. Th srucur of m mr s show h quo low.

4 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg m A [] Two mrcs c ddd or surcd f oh mrcs hv h sm sz. If w df mr, C, s h sum or dffrc of wo mrcs, A d B, w c wr hs sum or dffrc rms of h mrcs s follows. C A B possl oly f A d B hv h sm sz [5] Th compos of h C mr r smply h sum or dffrc of h compos of h wo mrcs g ddd or surcd. Thus for h mr sum or dffrc show quo [5], h compos of C r gv y h followg quo. C A B c, ;, m [6] Th produc of mr, A, wh sgl umr,, ylds scod mr whos sz s h sm s h of mr A. Ech compo of h w mr s h compo of h orgl mr,, mulpld y h umr. Th umr hs cs s usully clld sclr o dsgush from mr or mr compo. B A f, ;, m [7] W df wo spcl mrcs, h ull mr,, d h dy mr, I. Th ull mr s rrry shp my or my o squr mr whch ll h lms r zro. Th dy mr s squr mr whch ll h dgol rms r d h off-dgol rms r zro. Ths mrcs r somms wr s m or I o spcfy prculr sz for h ull or dy mr. Th ull mr d h dy mr r show low. I [8] A mr h hs h sm pr s h dy mr, u hs rms ohr h os o s prcpl dgol s clld dgol mr. Th grl rm for such mr s dδ, whr d s h dgol rm for row d δ s h Krockr dl; h lr s dfd such h δ = ulss =, whch cs δ =. A dgol mr s somms rprsd h followg form: D = dgd, d, d,,d; hs sys h D s dgol mr whos dgol compos r gv y d. m m m

5 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 5 D d d d d [9] W cll h dgol for whch h row d s h sm s h colum d, h m or prcpl dgol. Algorhms h umrcl lyss of dffrl quos ld o mrcs whos ozro rms l log dgols. For such mr, ll h ozro rms prculr dgol my rprsd y symols lk,-k or,+k. Dgols wh suscrps,-k or,+k r sd o l, rspcvly, low or ov h m dgol. If h rows d m colums mr, A, r rchgd, w wll hv w mr, B, wh m rows d colums. Th mr B s sd o h rspos of A, wr s A T. T B A f [, ;, m; A s m; B s m.] [] A mpl of orgl A mr d s rspos s show low. 6 T A 6 A [] Th rspos of produc of mrcs quls h produc of h rsposs of dvdul mrcs, wh h ordr rvrsd. Two mpls r show low. C you compl h hrd? T T T T T AB B A ABC C B A ABCD [] T T T Mrcs wh oly o row r clld row mrcs; mrcs wh oly o colum r clld colum mrcs. Alhough w c wr h lms of such mrcs wh wo suscrps, h suscrp of o for h sgl row or h sgl colum s usully o cludd. Th mpls low for h row mr, r, d h colum mr, c, show wo possl forms for h suscrps. I ch cs, h frs row or colum mr hs doul suscrp, such s for compo h sgl-row of row mr or for sgl colum of colum mr; h scod form hs h commoly usd sgl suscrp. Wh row d colum mrcs r usd formuls, such s h formul for h mulplco of wo mrcs show low quo [9], h hv wo mr suscrps, h frs form of h mrcs show low r mplcly usd o gv h mssg suscrp wh vlu of for h quo. Row d colum mrcs r clld row vcors or colum vcors wh hy r usd o rprs h compos of vcor. I hs os, w wll us uppr cs oldfc lrs such s A d B o rprs mrcs wh mor h o row or mor h o colum; w wll us lowr cs oldfc lrs such s or o rprs mrcs wh oly o row or oly o colum. W wll grlly rfr o hs mrcs s vcors.

6 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 6 r r r r r r r r m r m c c c c c c c c c [] Th rspos of colum mr s row mr; h rspos of row mr s colum mr. Ths s somms usd o wr colum mr h mddl of y syg, for mpl, h c = [ - 5] T. Mr Mulplco Th dfo of mr mulplco sms uusul wh courd for h frs m. Howvr, hs s orgs h rm of lr quo sysms. For smpl mpl, w cosdr hr wodmsol coord sysms. Th coords h frs sysm r d. Th coords for h scod sysm r y d y. Th hrd sysm hs coords z d z. Ech coord sysm s rld y coord rsformo gv y h followg rlos. y y z z y y y y [] W c o rloshp w h z-coord sysm d h -coord sysm y comg h vrous compos of quo [] o lm h y coords s follows. z z [ [ ] ] [ [ ] ] [5] W c rrrg hs rms o o s of quos smlr o hos quo [] h rls h z coord sysm o h -coord sysm. z z [ [ ] ] [ [ ] ] c c c c [6] W s h h coffcs c, for h w rsformo r rld o h coffcs for h prvous rsformos s follows. c c [ [ ] ] c c [ [ ] ] [7] Thr s grl form for ch c coffc quo [7]. Ech s sum of producs of wo rms. Th frs rm from ch produc s k vlu whos frs suscrp s h sm s h frs suscrp of h c coffc g compud. Th scod rm ch produc s k vlu whos scod suscrp s h sm s h scod suscrp of h c rm g compud. I ch kk produc, h scod suscrp k s h sm s h frs suscrp. From hs osrvos, w c wr grl quo for ch of h four coffcs quo [7] s follows.

7 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 7 c,;, k k [8] k Th dfo of mr mulplco s grlzo of h smpl mpl quo [8] o y grl szs of mrcs. I hs grl cs, w df h produc, C = AB, of wo mrcs, A wh rows d p colums, d B wh p rows d m colums y h followg quo. p m p p m k k m k C A B c,, ;,, [9] Thr r wo mpor ms o cosdr h formul for mr mulplco. Th frs s h ordr s mpor. Th produc AB s dffr from h produc BA. I fc, o of h producs my o possl. Th scod m s h d for comply w h frs d scod mr h AB produc. I ordr o o h produc AB h umr of colums A mus qul h umr of rows B. A smpl mpl of mr mulplco s show low. A 66 AB 6 6 B [] Mr mulplco s smpl o progrm. Th C++ cod for mulplyg wo mrcs s show low. Ths cod ssums h ll vrls hv proprly dclrd d lzd. Th cod uss h ovous oo o mplm quo [9]. Th rry compos r dod s [][k]. [k][] d c[][]. Th produc mr, C, hs h sm umr of rows,, s mr A d h sm umr of colums, m, s mr B. Th umr of colums A s qul o p, whch mus lso qul h umr of rows B. for = ; <= ; ++ for = ; <= m; ++ { c[][] =.; for k = ; k <= p; k++ c[][] += [][k] [k][]; } Th rms prmulply d posmulply r commoly usd o dc h ordr of h mrcs volvd mr mulplco. I h mr produc AB, w sy h B s prmulpld y A or h A s posmulpld y B. Alrvly, h rms lf mulpld d rgh mulpld r usd. I h AB produc, A s rgh mulpld y B d B s lf mulpld y A. Th sc cod srucur s h sm y lgug. Thr r hr sd loops. Th wo our loops covr ll possl comos of d o sur h ll h c compos r compud. Th r loop cod s h ypcl cod for summg umr of ms. C++ progrmmrs wll o h h loop dcs usd hs cod gor h fc h h mmum d for C++ rry s zro. Ths ws do dlrly for ll cod mpls hs os o provd smlr umrg for rry dcs h os d hos h cod.

8 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 8 W ow m how h coord rsformos h w usd ov o roduc mr mulplco c rprsd s mr quos. W c df mrcs, A, B, d C o rprs h coffcs h w usd our coord rsformo quos. B c C c A [] Th vrous coord prs c rprsd s colum mrcs s show low. y y y y y z z z c c z z [] Wh hs mr dfos, h wo ss of smulous lr quos show quo [] c rprsd y h followg pr of mr quos: y A d z By [] You c vrfy h h quos ov r corrc y pplyg h grl formul for mr mulplco quo [9] o h mr quos []. To do hs, you should us h dfos of A, B,, y, d z, provdd quos [] d []. If w com h mr quos [] o lm h y mr, w g h followg rsul. z By BA or z C wh C BA [] No h mporc of h ordr of mulplco. I grl, BA; s o qul o AB. Thr r wo css whr h ordr s o mpor. Ths r mulplco y ull mr, whch producs ull mr, d mulplco y dy mr, whch producs h orgl mr. A A d AI IA A [5] Alhough h ordr s o mpor hr, h cul dy d ull mrcs usd my dffr. W c rwr quos [5] o plcly show h rows d colums ch mr.. p A A m m I m m p m I A m A m m q A m q [6] By dfo h dy mr s squr mr. O sz spcfco for h dy mr, h umr of rows or h umr of colums, s s y h comply codo for mr mulplco. Oc hs s do, h ohr sz s s y h rqurm h I s squr. For h ull mrcs quo [6], h sz spcfcos, or m, mus mch h szs for h A mr. Alhough h sz spcfcos p d q, for h ull mrcs quo [6] r rrry, hy r usully k s p = m d q = o gv squr ull mr s h A produc. Smulous Lr Algrc Equos Th coord rsformo quos r smpl mpls of mor grl cs for smulous lr lgrc quos. I h grl cs, w c hv s of smulous quos h s wr s follows

9 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 9 m,, [7] W pc h wll-drmd prolm wll hv h umr of quos,, qul o h umr of ukows, m, u h grl cs, show ov, hs my dffr from m. W s h hs sysm of quos quo [7] c rprsd y mrcs A m, m, d, whr A hs h form show quo [], s h colum mr [,,, m] T d s h colum mr [,,, ] T. Wh hs dfos, quo [7] s h sm s h grl quo for mr mulplco show quo [9]. Rcll h w hv omd h scod suscrp, whch s o, o h compos of d. Th sysm of quos show quo [7] s wr, mr form, quo [8], low. A = [8] Ths mrcs r wr ou dl low. Hr h colum mr,, pprs o hv mor rows h h coffc mr, A. Ths s do o mphsz h oo h m my dffr from grl. Of cours, m my qul o or lss h rhr h grr h s mpld h mrcs show h quo low. m m m m m [9] I ordr o solv s of smulous lr quos w us procss h rplcs quos h s y quvl quos. W c rplc y quo h s y lr como of ohr quos whou chgg h soluo of h sysm of quos. For mpl, cosdr h smpl s of wo quos wh wo ukows 7 5 [] You c cofrm h = d = s soluo o hs s of quos. To solv hs s of quos w c rplc h scod quo y w quo, whch s lr como of h wo quos whou chgg h soluo. Th prculr como w sk s o h wll lm. W c do hs y surcg h frs quo, mulpld y 7/, from h scod quo o o h followg pr of quos, whch s quvl o h orgl s quo [] []

10 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg W c rdly solv h scod quo o fd =, d susu hs vlu of o h frs quo o fd h = [ 5]/ =. Th grl procss for solvg h sysm of quos rprsd y quos [7], [8], or [9], kow s Guss lmo, s smlr o h o us show. I rqurs srs of opros o h coffcs d o produc s of quos wh h form show quo [], low, whou chgg h soluo of h l prolm. [] Th sc rul h Guss lmo procss s h w c us lr como of wo quos o rplc o of hos quos, whou chgg h soluo o h prolm. Ths s h procss h w usd ov gog from h s of quos [] o h s of quos []. Boh ss of quos r quvl h ss h oh ss of quos gv h sm swrs for d. Howvr, h scod s of quos c drcly solvd for ll h ukows. Th rvsd coffc mr quo [] s clld uppr rgulr mr. Th oly ozro rms r o or ov h prcpl dgol. Th sm opros h r usd o o h rvsd coffc mr r usd o o h rvsd rgh-hd-sd mr. Th rvsd A d mrcs r od srs of sps. I h frs sp, h coffcs r lmd from ll quos cp h frs o. Ths s do y h followg rplcm opros o h coffcs quos o. Th rplcm oo from compur progrmmg s usd hr o dc h old vlu of s g rplcd y h rsuls of clculo. Ths vods h d o us mhmcl oo h would rqur spr symols for h old vlu d h w vlu of., d, [] Afr quo [] s ppld o ll rows low h frs row, h oly ozro coffc s h frs quo rprsd y h frs row of h mr. You c cofrm h hs wll s = for >. You c lso pply h formul [] o quo [] o s h h rsul s quo []. Th lmo procss s ppld o mk h coffcs o ll quos low h scod quo zro., d, [] Equo [] hs h sm form s quo []; oly h srg pos for h row d colum opros r dffr. Th procss dscrd y quos [] d [] cous ul h form show quo [] s od. From quo [], h vrous vlus of c foud y ck susuo. W c smply fd s β/α. Th rmg vlus of r foud rvrs ordr y h followg quo.

11 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg,,, [5] Wh w r solvg for, ll prvous vlus of rqurd h summo r kow. Th C++ cod low shows smplfd vrso of how h Guss lmo mhod s ppld o h soluo of quos. As prvous cod mpls, ll d vlus r ssumd o proprly dclrd d lzd. Th umr of quos s qul o h umr of ukows,. Th row h s surcd from ll rows low s clld h pvo row. Th m our loop h frs pr of h cod uss h vrl, pvo, o rprs hs row. Th cod cuo s smplfd y ugmg h mr so h,+ =. Ths llows h cod o procd whou spr cosdro of smlr opros o h A d mr compos. // ugm mr wh vlus for row = ; row <=; row++ [row][+] = [row]; // g uppr rgulr rry for pvo = ; pvo < ; pvo++ for row = pvo+; row <= ; row++ for colum = row+; colum <= +; colum++ [row][colum] -= [row][pvo] [pvo] [colum] / [pvo][pvo]; // Uppr rgulr mr compl; g vlus for row = ; row <= ; row-- { [row] = [row][+]; for colum = ; colum < row; colum-- [row] -= [row][colum] [colum]; [row] /= [row][row]; } Th procss ould ov for h soluo of s of smulous quos s kow s h Guss lmo procdur. Alrv procdurs such s h Guss-Jord mhod d LU dcomposo, work smlr mr. Thy produc uppr rgulr mr or dgol mr h s h usd o solv for h vlus of rvrs ordr. Mr Rk Drms Esc d Uquss of Soluos If h soluo procss ould ov s usd o cr mrcs, my o possl o o soluo. Cosdr h wo ss of quos show low d [6] I h s of quos o h lf, h scod quo s smply wc h frs quo. If w mulply h frs quo y wo d surc from h scod quo, w g h rsul h =. Thus, h Acul cod would hv o ccou for h possly h h sysm of quos mgh o hv soluo. I would lso us dffr opros o rduc roud-off rror. Ths mpl cous h prcc usd prvously of srg h rry suscrps d dg hm o coss wh h oo quos hs os. Typcl C++ cod srs h rry suscrps.

12 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg scod quo gvs us o w formo o h rloshp w d. W sy h hs sysm of quos hs f umr of soluos. Ay vlu of =.6.6 wll ssfy oh quos. Th scod s of quos hs o soluos. If w mulply h frs quo y wo d surc from h scod quo, w hv h rsul h =! Thus, hs scod s of quos s compl d dos o hv soluo. 5 Ths smpl mpl c grlzd o dscuss h sc d uquss of soluos for h grl s of quos. If w crry ou h soluo procss ould ov o form uppr rgulr mr, w my hv h rsul h o or mor of h fl rows h coffc mr s ll zro. Ths ms h w co o uqu soluo. Such cs s clld sgulr mr. Th rk of mr s formlly dfd s h umr of lrly dpd rows mr. Ths c show o qul o h umr of lrly dpd colums. Th prccl drmo of rk s sd o h Guss lmo procss ould ov. If h fl mr h lmo procss s mr wh rows of whch zro rows co ll zros, h rk of h mr s zro. Ths rk s h sm for oh h orgl mr d h uppr-rgulr mr cus h Guss lmo opros do o chg h mr rk. Th A mr for oh ss of quos quo [6] hs oly o lrly dpd row, hus s rk s o. Th uppr rgulr form h rsuls wh mr s sd for rk s somms clld h row-chlo form. Somms hs form ch row s dvdd y h dgol lm o h row so h ll h dgol lms r o. Th wo mrcs quo [7] low hv plcd row-chlo form y usg Guss lmo o h orgl mrcs. C you drm h rk of h orgl mrcs for lookg h swrs low? Th mr o h lf of quo [7] hs four rows h r o ll zro; hus, s rk s four. Th o o h rgh hs s rows h r o ll zro, hus s rk s s. Ths rk-s mr hs gh colums. Bcus h umr of lrly dpd colums d h umr of lrly dpd rows r oh h sm s h rk of s, w kow h hs gh colums wll rld y wo dffr lr quos [7] Th sc d uquss of soluos r dfd rms of h rk of h ugmd mr, [A,]. Ths s h mr whch h rgh-hd sd colum mr,, s ddd s r colum h A mr. Ths ugmd mr s show low for h grl cs of quos d m ukows. Th quos m h hr r rows h mr. Th m ukows gv m + colums o h ugmd mr. 5 Ths rsul hs gomrc rpro. Wh w hv wo smulous lr lgrc quos, w c plo ch quo spc. Th soluo o h pr of quos s locd h po whr oh quos rsc. If w dd hs for h lf s of quos [6], w would oly hv sgl l.

13 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg [, ] m A [8] Th sc d uquss of soluos o A = s sd low whou proof. If h rk of h orgl mr, A, quls h rk of h ugmd mr, [A,], d quls h umr of ukows, m, hr s uqu soluo o h mr quo, A =. If h rk of h orgl mr, A, quls h rk of h ugmd mr, [A,], u s lss h h umr of ukows, m, hr r f umr of soluos o h mr quo, A =. If h rk of h orgl mr, A, s o qul h rk of h ugmd mr, [A,], hr s o soluo o h mr quo, A =. W c s h hs sms r coss wh h mpls quo [6]. A forml proof of hs sms s gv lr lgr s. Ths gudls for h sc d uquss of soluos o smulous lr quos r llusrd h hr ss of quos show low. Ech quo s hs hr quos hr ukows. Th orgl quo s, show h frs colum, s covrd o uppr rgulr form h scod colum. W s h h frs s hs uqu soluo. Th scod d hrd ss do o hv uqu soluo; howvr, hr s dffrc w hs wo. Th scod s hs f umr of soluos. For y vlu, h w pck for w c drm vlu of d h s coss wh h orgl s of quos. Howvr, for h hrd s of quos, h uppr rgulr form gvs coss hrd quo. Thus, hs s of quos hs o soluo. m m m S I S II S III Orgl Equo S Uppr Trgulr Form Soluos No Soluo Ths hr ss of quos r show rms of hr A d ugmd [A ] mrcs h l low. W s h h s of quos h l ov corrspods o h d h ugmd mr. Th frs s of quos hs rk A = rk [A ] =, h umr of ukows. W hv lrdy s h hs provds h uqu soluo ov. Th scod s of quos hs rk A = rk [A ] =, lss h h umr of ukows. Ths ms h w hv f umr of soluos. Ag, hs

14 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg corrspods o h rsul ov. Flly, h hrd cs low hs rk A =, u rk [A ] =. Ths dffrc rk shows h hr r o soluos. Orgl Mrcs Row-Echlo Form Rk A [A ] A [A ] A [A ] S I S II S III Thr s o fl cs o cosdr; h s h cs of homogous quos, whr h mr s ll zros. If hr r quos d h rk of h coffc mr s h h oly soluo o h s of quos s h ll =. Ths s clld h rvl soluo. Howvr, f h rk s lss h, s possl o hv soluo whch ll h r o zro. Howvr, such soluo s o uqu. Cosdr h wo ss of homogous quos show low. Ech s of quos hs rgh-hd sd h s ll zros. Th wo quo ss r dcl cp for h coffc of h rm h frs quo d [9] If w crry ou h usul soluo procss o cr uppr rgulr mr for hs wo ss of quos, w o h followg rsuls d [] For h s of quos o h rgh, h rk of h coffc mr s h sm s h umr of quos. Hr w hv uqu soluo whch ll of h =. Th rk of h coffc mr for h quos o h lf s lss h h umr of quos. I hs cs, w hv f umr of soluos. If w pck = α, rrry cos, w c ssfy ll hr quos f = α/ d = α α/ = α/. O of h f soluos, wh =, s h rvl soluo whr ll =. 6 6 You should mk sur h you c plc h orgl ss of quos [9] d h uppr rgulr forms [] o A d ugmd [A ] mr d show h oh ss of quos hv rk A = rk [A ]. Do oh

15 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 5 Drms A drm s sgl umrcl vlu h c compud for squr rry. Th vlus of drms ply horcl rol mr lyss d c usd for clculos o smll mrcs. For mrcs whos rk s grr h or, lrv clculo mhods r usd plc of drms. Vrous oos r vll for drm. If A s mr, h D A s h drm for h coffcs mr. Th drm for rry of umrs h looks lk mr c wr usg h solu vlu sg, <rry>, sd of h rcks, [<rry>], h w hv usg for mr coffcs. Th vrous oos r show low for rry. For hs rry, h formul for h drm, whch s show s h fl pr of quo [], s prculrly smpl. A D D A [] For rry, h drm s mor compl. D [] Th grl quo for compug drm s gv rms of mors or cofcors of drm. Th mor, M of drm s h smllr drm h rsuls f row d colum r lmd from h orgl drm. Th cofcor, C, quls - + M. For mpl, f w sr wh drm, such s h o show quo [] w c df possl mors d cofcors. Four of hs r show low: C C M M C C M M [] Th drm of mr c wr rms of s mors or cofcors s follows. D A M M C C [] No h h sum s k ovr y o row or ovr y o colum. I pplyg hs formul, o sks rows or colums wh lrg umr of zros o smplfy h clculo of h drm. W c show ss of quos produc A mrcs wh h sm rk? Wh r h rks of h A d [A ] mrcs for h wo ss of quos?

16 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 6 h hs quo s coss wh h rsuls gv prvously for h drms of d rrys. Applyg quo [] o h hrd row of rry gvs h followg rsul. D C C C C C [5] W could hv ppld quo [] o y of h hr rows or y of h hr colums o compu h drm. I chos o us h hrd row sc h cssry cofcors c foud quo []. If w us quo [] o pd h cofcors [] d pply hos rsuls o quo [5], w o h followg rsul. D A M M M [6] Th fl rsul, fr som rrrgm, s h sm s h o quo []. Two ruls ou drms r ppr from quo [5]: A drm s zro f y row or y colum cos ll zros. If o row or o colum of drm s mulpld y cos, k, h vlu of h drm s mulpld y h sm cos. No h mplco for mrcs: f mr s mulpld y cos, k, h ch mr lm s mulpld y k. If A s mr, DkA = k DA. Addol ruls for d proprs of drms r sd low whou proof. If o row or o colum of drm s rplcd y lr como of h row or colum wh ohr row or colum, h vlu of h drm s o chgd. Ths ms h h opros of h Guss lmo procss do o chg h drm of mr. If wo rows or wo colums of drm r lrly dpd h vlu of h drm s zro. Th drm of h produc of wo mrcs, A d B s h produc of h drms of h dvdul mrcs: DAB = DA DB. Th drm of rsposd mr s h sm s h drm of h orgl mr: DA T = DA. If w pply h colum pso of quo [5] o uppr rgulr mr, A, w fd h D A = A, sc h rm s h oly rm h frs colum. W c pply quo [5] rpdly o h cofcors. Ech pplco shows h h drm s smply h w rm h uppr lf of h rry ms s cofcor. Coug hs fsho w s h h drm of uppr rgulr mr s smply h produc of h dgol rms. W c com hs rsul wh h fc od ov h h opros of h Guss lmo procss do o chg h drm of mr o dvlop prccl for compug drms of y mr. Apply Guss lmo o g h mr uppr rgulr form h h drm of oh h orgl mr d h o uppr rgulr form s smply h produc of h dgol lms.

17 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 7 As mpl, cosdr h mrcs from S h l o pg. Th orgl mr ws ; s uppr rgulr form ws 9. W c rdly compu h drm s h produc 5.5 =. You c show h h sm vlu s od y h covol formul for h vluo of h orgl drm. Drms r o usd orml umrcl clculos. Howvr, f you d o fd h umrcl vlu for lrg drm, h procss ould ov s h mos drc umrcl pproch. Crmr s rul gvs h soluo o sysm of lr quos rms of drms. Ths pproch s vr usd cp for vry smll umrs of quos, ypclly wo or hr. Accordg o Crmr s rul h soluo for prculr ukow s h ro of wo drms. Th drm h domor uss ll h usul mr coffcs,. Th drm h umror cosss of h coffcs cp o colum. Wh w r solvg for w rplc colum h coffcs y h rgh-hd-sd mr coffcs,. For s of hr quos hr ukows, Crmr s rul would gv h soluos show quo [7]. Crmr s rul llows us o fd lycl prsso for h soluo of s of quos, d s somms usd o solv smll ss of quos or. Howvr, s vr usd for umrcl clculos of lrgr sysms cus s rmly m cosumg. [7] Drms r lso rld o rk. A rry whch h rows r o lrly dpd wll hv zro drm. As mpl of hs cosdr h lf-hd s of quos from quo [9]. Rcll h h coffcs for h s of hr quos hd rk of wo cus h quos wr o lrly dpd. Wh w vlu h drm for hs rry low, usg quo [] for h drm of rry, w fd h h drm s zro [8] Ths gvs us ohr pproch o drmg wh s of quos wh ll zros o h rgh-hd sd hs soluo ohr h h smpl o h ll r zro. Ths codo s h h drm of h coffc mr s zro. If DA = h soluo o A =, whr cos ll zros, h dos o hv ll = s possl. Thr r cully f umr of such soluos. Ths soluos dffr y rrry mulplr. W wll us hs d low wh cosdrg mr gvlus.

18 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 8 Ivrs of Mr W hv dfd opros for ddg, surcg d mulplyg mrcs. Mr vrso s h mr log of dvso. For squr mr, A, w df mr vrs, A -, y h followg quo. AA [9] A A I If w hv mr quo, A =, w c, prcpl, solv hs quo y prmulplyg oh sds of h quo y A -. Ths gvs h followg rsul. If A, A A A I A A [5] Th vrous sps quo [5] us h dfo, quo [9], h h produc of mr d s vrs s h dy mr d h dfo, quo [5] h h produc of y mr wh h dy mr s h orgl mr. Alhough h rsul h = A - my wr s h soluo o h orgl quo, h cul soluo of mr quos lk A = s do y mhods ohr h h drc clculo of h vrs. I s o lwys possl o fd h vrs. A squr mr h hs o vrs s clld sgulr mr. I s usully o cssry o fd h vrs of mr. If cssry, you c fd umrcl vlu of h vrs y h sm procss usd o solv smulous lr lgrc quos. To udrsd how hs s do, w df scod mr, B, s A -. Th, y h dfo of vrs w hv h followg quo. Equo [5] shows h mrcs volvd hs quo. B A If, AB I [5] [5] W hv form smlr o h usul prolm of solvg s of quos. Th coffc mr, A, s h sm, u w hv rgh-hd sd colums of kow vlus. Ech of hs colums of kow vlus corrspods o o colum of ukows h B mr h s A -. If w us our usul procss for solvg A =, wh, for mpl, = [ ] T, w wll o h frs colum of B = A -. Rpg h procss for smlr colums, whch r ll zros cp for row k gvs us colum k of h vrs. For mpl, quo [5] shows h soluo for h scod colum of B = A -.

19 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 9 [5] Bcus h opros for solvg s of smulous lr quos r sd o h A mr oly, h soluo for h vrs s cully do smulously for ll colums. A lycl prsso for h vrs c od rms of h cofcors dscussd h sco o drms. W cou o df B = A - ; h compos of h vrs,, r h gv rms of h mors or cofcors, C, of h orgl A mr d s drm. If C M B A, [5] D A D A Th smpls pplco of hs quo s o mr. For such mr, h coffcs of B = A - r gv y h followg quos. M D A M D A D A D A M D A D A M D A D A [55] Comg h rsuls of quo [55] wh quo [] for drm, gvs h followg rsul for h vrs of mr. [56] You c sly show h hs s corrc y mulplyg h orgl mr y s vrs. You wll o u mr y hr mulplco: AA - or A - A. Th sm procss c usd o fd h vrs of mr; h rsul s show low: [57]

20 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg Equos [56] d [57] show h vlu of drms provdg lycl soluos o vrss. Alhough drms r vlul such css y us of drms should vodd umrcl work. Th grl rul for h vrs of mr produc d h vrss of h dvdul mrcs s smlr o h sm quo for h rspos of mr produc d h produc of h rsposs of h dvdul mrcs. Ths rlo s show low. AB B A ABC C B A ABCD [58] Mr Egvlus d Egvcors If squr mr c prmulply colum vcor d rur h orgl colum vcor mulpld y sclr, h sclr s sd o gvlu of h mr d h colum vcor s clld gvcor. I h followg quo, h sclr, λ, s gvlu of h mr A, d s gvcor. W c us h dy mr o rwr hs quo s follows. A [59] [ A I ] [6] m [6] As dscussd ov h scos o smulous lr quos d drms, quo [6] hs h soluo h ll vlus of r zro. I my hv ozro soluo f h drm of h coffc mr s zro. Th s, D[ A [6] I] D From h grl prsso for drm, w s h o compo of h fl prsso for drm of y sz s h produc of ll lms o h prcpl dgol. I quo [6] hs rm wll gv h ordr polyoml for our mr. Ths h ordr polyoml s kow s h chrcrsc quo of h mr. Ths chrcrsc quo c solvd for vlus of, o ll

21 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg of whch my dsc. For wo-y-wo mr, sg D[A Iλ]= gvs h followg qudrc quo. [6] W c solv h qudrc quo [6] o g wo roos h gv us h wo possl gvlus: [6] Ech gvlu wll hv s ow gvcor. Ech gvcor s foud y h soluo of quo [57]. If w do h gvcors s d, h compos of gvcor my wr s d. Accordgly, w hv o solv h s of quos show low wo ms oc for d oc for. [65] Ag, h soluo s o uqu. Ay s of vlus, mulpld y rrry cos, wll ssfy hs s of quos. For smplcy w pck = α. W hv wo possl rsuls for h gvcor compo, dpdg o whch quo w us. [66] Thr ppr o wo dffr soluos for, dpdg o h us of h frs or scod quo o g hs gvcor compo. Howvr, qug hs wo vlus for, wll lm h rrry cos, α, d o quo [6] h w solvd for λ. Thus h wo possl prssos for quo [66] wll rsul h sm vlu. As umrcl mpl, cosdr h drmo of h gvlus d gvcors for h mr, A 5. You c fd h swr usg quos [65] d [66]. Howvr, w wll oul h r soluo procss s mpl of fdg gvlus d gvcors for lrgr sysms. Solvg h quo D[A I] for hs mr gvs h followg rsul. 5 D [ A I] 5. Th roos o hs quo r = d =. Hr w hv usd h umrg covo h h hghs gvlus hs h lows d. W ow susu ch gvlus o h quo A I =, d solv for h compos of ch gvcor. For h frs gvcor w o. 5 [67] W s h h ls quo rsuls =, whch gvs us o usful formo. Sc w kow h h homogous quo s hs f umr of soluos, w pck rrry vlu,, for. Wh

22 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg hs vlu, h frs quo gvs us h rsul h = 5. Thus our frs gvcor, = [5 ] T. W c pply h sm procdur o fd h scod gvcor. 5 [68] Hr oh quos ll us h mus zro. Howvr, hr s o formo ou. W coclud h hs mus rrry quy h w wll cll. Ths gvs our scod gvcor, = [ ] T. W c vrfy our soluo for gvlus d gvcors y showg h hy ssfy h dfg quo, A =. A [69] A 5 5 [7] Th clculos ov show h h dfo of gvlus d gvcors s ssfd rgrdlss of our chocs for d. Ths s grl rsul. W r lwys fr o choos o compo of h gvcor. Howvr, h rmg compos wll s. Typclly h gvcor compos r chos o gv smpl prsso for h gvcor., o whch ll h compos r grs or smpl frcos or u vcor. 7 I h mpl of h wo-y-wo mr usd ov, w could prss h gvcors w foud y of h wys show mmdly low. Th ls prsso show for ch gvcor s u vcor. No h h wo gvcors r o orhogol hs mpl [7] Mr Trsformos Usg Egvcors I s possl o us h gvlus d gvcors o rsform h orgl A mr o dgol mr. To do hs w df mr, X, whos colums r h dffr gvcors. Th s X = [,,, ]. W lso df dgol mr, Λ, whos lms r h gvcors;.., Λ = [λδ]. Th mr produc, AX c vwd s h produc of A wh ch gvcor. Th s AX = [A, A, A, A] = [λ, λ, λ, λ]. W c us hs rsul o show h AX = X Λ. To do hs, w m h mr produc XΛ low, whr w rgrd h suscrp h dfs h prculr gvcor s colum d. 7 A u vcor, u, s o for whch h wo orm, u = u =. If w hv vcor, v, h s o u vcor w c covr o u vcor y dvdg ch compo y v. Ths would gv h compos of h w u vcor y h followg quo: u v v k

23 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg XΛ [7] Crryg ou h dcd mr mulplco gvs h followg rsul. AX XΛ [7] Equo [7] gvs h dsrd rsul h XΛ = AX; w c prmulply hs quo y X - o o h followg rsul. AX X Λ [7] Th s, w c us h mr crd y usg h gvcors s mr colums o produc dgol mr from h squr mr, A. Th ozro compos of h dgol mr r h gvlus of h A mr. Ths yp of rsformo s mpor my dvcd pplcos of mr hory. A mpl of hs, h us of mr gvlus h soluo of sysm of ordry dffrl quos, s show h sco. I h mpl srd o pg, w foud h gvlus, = d =, d h gvcors = 5 d = for h mr, 5 A. Usg hs gvcors, w fd X = 5. W us quo [56] o g X - = 5 5 Susug X -, A, d X o quo [7] gvs h followg rsul AX X

24 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg [75] 5 Ths X - AX produc producs h pcd rsul: dgol mr wh h gvlus of A o h dgol. W lso s h h rsul dos o dpd o h rrry mulplcv cos usd ch gvcor. Smlry rsformos A mpor cocp h compuo of mr gvlus s h of smlry rsformo. For squr mr, A, h rsformo, B = P - AP, whr P s y vrl squr mr, producs w mr, B h hs h sm gvlus s A. W c prov hs y srg wh h gvlu quo for B, B =, d susug h rsformo B = P - AP. B = P - AP = [76] If w prmulply ch sd of h ls quo y P, w c mpul h rsul s follows. PP - AP = IAP = AP = P [77] Sc s sclr, w c wr P = P, so h h ls quo [77] coms gvlu quo for A. AP = P [78] Equo [78] lls us h w c mulply h vcor, P, y h mr, A, d o h sm vcor, mulpld y. Ths s gvlu/gvcor quo whr h gvlu for h A mr s h sm s h gvlu for h B mr. Th gvcors of h A d B mrcs, rld y smlry rsformo, B = P - AP, ssfy h followg rloshps: A = P B or, quvlly, B = B = P - A. Applco of Mr Egvlus d Egvcors o Sysm of Dffrl Equos Ths rsform my usd h soluo of smulous dffrl quos. A grl sysm of lr, frs-ordr dffrl quos for vrls, y, c wr s follows: dy d N y r, N [79] I hs quo, h vlus of r cos. If w df y d r s colum mrcs, w c rwr hs sysm of dffrl quos s mr quo: If y s mr, wh compos y, h h drvv dy/d s mr whos compos r dy/d.

25 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 5 dy Ay r d [8] W c us h mr, X, whos colums r gvcors of h A mr o df w s of dpd vrls, s whos compos df h colum mr, s, y h followg quo. y Xs or s X y [8] Wh hs dfo, h orgl mr dffrl quo [8] c wr s dffrl quo s. dxs AXs r d ds X AXs r d [8] Th scod quo s possl cus w hv ssumd h ll h vlus of r cos;.., hy do o dpd o m. Ths ms h h gvcors h X mr wll coss s wll. If w prmulply h scod quo [8] y X -, w o h followg rsul, usg quo [7] o rplc X - AX y h gvlu mr, Λ. X ds X X d AXs X r ds I Λs X d r [8] If w df w rgh-hd sd colum mr, p = X - r our sysm of dffrl quos c wr s follows. ds d Λs p [8] Sc Λ s dgol mr, h dffrl quo for ch compo, s dpds oly o s d p. I dos o dpd o ohr compos of h s rry. Thus, w hv s of dpd sclr dffrl quo o solv. ds d s p [85] Ths dffrl quo hs h form of h grl frs-ordr dffrl quo, d d f f [86] Th soluo o hs grl quo s show low. fd fd f d C [87] Th cos, C, s foud from h l codo o h dpd vrl. Equo [85] c plcd h grl form of quo [86] f w df f = λ d f = p. Sc λ s o fuco of m, w c wr h soluo o quo [85], rplcg y s s h dpd vrl, s follows.

26 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 6 q C p d C C p d s [88] Hr w hv dfd q, whch dpds o h prolm-spcfc vlu for p = X - r. d p q [89] If p s cos, w hv h followg rsul for q. p p p p d p p d q cos [9] I hs cs, h soluo for y coms p C y [9] If w kow h l vlus of y =, dod s y, w could fd h corrspodg l vlus of s from h dfo of s quo [8]. If w do h l vlus of s s s, w would fd hs from h l y vlus s follows. Xs y y X s [9] Wh hs dfo of s, wh compos, s, w c solv for h cos, C, quo [88] s follows. q s C q C s [9] Wh hs vlu for C, h soluo o our dffrl quo coms: q q s s [9] For h spcl cs whr ll h p r cos w hv q = p/i, d sc q s o fuco of m hs cs, q = p/i. p p s s [95] W c rwr quo [9] s mr quo f w df h mr E s h followg dgol mr. N ] [ E [96]

27 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 7 I ordr o rwr quo [9] or quo [95] s mr quo, w hv o rrrg h ordr of h pol rms o gv h corrc rsuls from mr mulplco. s s q q s q q [97] s Es q q [98] If w susu h dfos of s from quo [8] d h dfo of s from quo [9] w o mr quo for h orgl y vrl. s q X y Es q q E X y q [99] Mulplyg hs quo y X gvs h soluo for y s follows. XE X y q y q [] Equo [95] gvs h soluo for cos p. Ths quo cos h rm p/ whch c wr s h produc of wo mrcs s show low. Ths quo uss h rsul h h vrs of dgol mr [] s h dgol mr [km/k]. 8 p p p p N N p p p N pn p [] Usg hs rsul d h sps h ld o quo [99], w c rwr quo [95] s show low. E p p Es I E p s Es [] If w susu h dfos of s from quo [8] d h dfo of s from quo [9] w o mr quo for h orgl y vrl. X I E p y X EXy [] I h smpls cs whr ll h rgh-hd-sd rms r h orgl quo [79] r zro, w wll hv p = d h soluo coms. y X EXy [] Ths quo lls us h h hvor of h soluo dpds o pol rms whos m coffcs r h gvlus of h mr from h orgl s of dffrl quos. 8 Th usul formul for h produc of wo mrcs, cm = kkkm gvs h followg rsul for h produc of h orgl mr d h proposd vrs: cm = kkkm/m = m. Hr w us h fc h h produc kkm s zro ulss = k d m = k. Th rsul cm = dm s h rqurd u mr for h produc of mr d s vrs.

28 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 8 Spcl mrcs d qudrc forms A s of colum mrcs,, s sd o orhogol f sclr produc of gv mr wh y ohr mr, cp slf, s zro. W c wr h dfo of orhogol s of mrcs s follows: Th s T d s orhogol f [5] If ll h vlus of d r o, h s of mrcs s orhoorml. Ay s of orhogol mrcs my covrd o orhoorml s y dvdg ch mr, y d. A orhogol mr s o for whch h vrs of h mr quls s rspos. Th s, h mr, A, s orhogol f A - = A T. O cosquc of hs dfo s h h rows d colums of orhogol mr form orhogol s of row or colum mrcs. Mr lms my compl umrs s wll s rl umrs. W hv lrdy dfd h rspos, B = A T, of mr, A, s o for whch =. For compl mrcs.., mrcs wh compl compos, w df h do mr, A, s h rspos of s compl coug. T T A A B A [6] W hv usd h oo h s h compl coug of ; som s us h oo ā o do h compl coug 9 of. Th mr, A, s od from h mr A, y rplcg ch compo,, y s rspos compl coug,. Mr oo vrs mog sourcs; som uhors us h oo A or A H for do mr. Ths uhors h us h oo, Ā, for h compl coug of mr A. A ury mr s grlzo of h orhogol mr for compl-vlud mrcs. A ury mr, U, s o for whch h do, U, quls h vrs, U -. A slf-do mr s o for whch A = A. Such mr s lso clld Hrm mr. A rl symmrc mr s slf-do or Hrm mr. A orml mr s dfd s o for whch h produc of mr wh s do dos o dpd o h ordr of h mulplco. Ths ms h AA = A A, f A s orml mr. Boh Hrm mrcs d rl symmrc mrcs r orml. Th mpor fur of orml mrcs s h hr gvcors form compl orhogol s. Ths ms h h X mr, dscrd for quo [7] d dfd mplcly h quo, wll hv vrs. I ddo, s smpl o drm h vrs of h gvcor mr, cus mus orhogol mr. From h dfo of orhogol mrcs hs ms h X - = X T. My grg pplcos yld symmrc mrcs, whch r gurd o provd orhogol gvcor mr. 9 If z s compl umr whos rl pr s d whos mgry pr s y, w wr z = + y whr = -. W c lso wr z = r θ, whr r = + y, d θ = - y/. Th compl coug, z = z = y = r -θ Th produc of compl umr wh s compl coug quls h mgud of h compl umr: z = z z = zz = r = + y.

29 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 9 To llusr hs mpor rsul for orml mrcs, cosdr h followg prolm h drms h gvcors usd h rsformo X - AX for h followg rl symmrc mr: A. W frs o h gvlus y solvg h quo h DA I =. Ths quo gvs D A I [7] Dog h dcd lgr gvs h followg cuc quo for [8] Th gol sk ool of Ecl ws usd o fd h gvlus of hs quo. Th hr gvlus r = , = , d = You c vrfy h hs vlus r h soluos o h gvlu quo. Th gvcors r foud y solvg h mr quo A I =. For h frs gvlu, w hv o solv h followg mr quo [9] Applyg Guss lmo o hs sysm of quos gvs h followg rsul [] As w pcd, our gvcor quo hs f soluos. If w pck vlu of o for, w o = [,5869]/ =.979 d = [.979] / = Th orm of hs gvcor s h squr roo of h followg sum: = Thus h orm of hs frs gvcor, =.5768 / =.655. If w dvd ch compo y hs orm, w wll o h followg ormlzd gvcor = [ ] T. You should l o vrfy h h wo orm of hs gvcor s o d h ssfs h quo h A I =. W c rp hs procss for h ohr wo gvcors d h w c o h X mr whch ch colum s o of h gvcors. Th vrs of hs mr s lso show low. Ths ws foud usg h MINVERSE fuco of Ecl.

30 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg X X [] W s h h vrs mr hs cs s h sm s h rspos of h X mr. Ths cofrms h h s of gvcors s orhogol. You should lso l o show h y pr of gvcors hs do produc r produc h s zro. Ths mpl llusrs h fc h symmrc mr producs orhogol s of gvcors. Ths grly smplfs h X - AX rsformo procss sc X - = X T for orhogol mr. A posv df mr s Hrm mr h ssfs h followg rloshp for y colum mr colum vcor h s o. You should kp md h Hrm mr whos compos r rl s smply symmrc mr. Posv Df A : A [] If ll h compos of r rl, s smply T. A posv sm-df mr s o for whch h grr-h sg s rplcd y grr h or qul o sg. Posv Smdf A: A [] Th produc A s llusrd low. A [] Tkg h A produc gvs h followg rmd rsul.

31 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg A [5] Th fl produc of row mr ms colum mr ylds y mr, whch s sslly sclr. Ths sclr s gv y h followg mr mulplco. A [6] W c prss h rsul of hs mr mulplco y sgl sclr, usg summo oo., A [7] W s h h rgh hd sd s sum of pur qudrc rms, d md qudrc rms,. Ech pur qudrc rm,, occurs oly oc d s mulpld. As usul, h produc of umr, wh s compl coug gvs rl umr h s h mgud of h orgl compl umr. Also, for Hrm mr, whr =, h lms o h prcpl dgol,, mus rl umrs for h grl dfo of Hrm mr o pply. A gv md qudrc rm occurs wc: h form, d oc h form. Th ddo of hs wo rms c smplfd y h us of h sc rloshp for Hrm mr h =. As show low, hs wo rms ld o h sum of wo ms h rl vlu of h ch dvdul rm. R [8] Usg hs rloshp w c rduc h umr of rms h fl summo of quo [7] y summg oly ovr vlus of h r grr h. Ths s sd o h rsul h h sum of compl umr, z = + y, d s compl coug, z = y, quls, wo ms h rl compo. Smlrly, z z = y, ms h mgry compo.

32 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg A R [9] If h mrcs A d r rl w c wr quo [9] s follows. T A [] If A s posv df, w r ssurd h y grl qudrc form lk h os quos [9] or [] wll posv. If A s posv sm-df w kow h h sums wll grr h or qul o zro. W c show h mr s posv df or sm-df y fdg ll s gvlus. A Hrm mr s posv df f ll s gvlus r posv; s posv sm-df f ll s gvlus r zro or posv. Vcor spcs, orms, d r producs Svrl cocps h w roducd wh vcors c grlzd o ohr rs of grg lyss. Ths grl pcur s clld src vcor spc. Th m src s usd cus h hgs h w rprs such spcs my o h sm s h rdol vcors w r usd o mchcs. Howvr, rdol vcors r o m h s rprsd vcor spc. W rcogz h vcors r usully rprsd y wo or hr compos wo- or hrdmsol spc. W df do produc for vcors d w df h mgud of h vcor s h sum of h squrs of s compos. W kow h w co rprs hr-dmsol vcor wo-dmsol spc, d w kow h w c rprs vcor mor h o wy y usg dffr coord sysm. Wh w rprs vcor, w lk o us orhogol coord sysm s h sc wy o rprs h vcor. I hs cs, w kow h h do producs of u vcors dffr coord drcos r zro. All h ds hs prgrph should fmlr o you for vcors rprsg forc, vlocy, cclro d h lk h you hv courd your grg d physcs courss. W w o grlz hs ds o sysms h c hv y umr of compos, o us wo or hr. I ddo, w w o cosdr fucos s wll s umrs s h compos. W lso w o cosdr h possly h h lms h w rprs my hv compl vlus. Th s quy, z, my rprsd s h sum of rl pr,, d mgry pr, y; w wr hs s z = + y, whr =. W df h complm of hs compl umr wh h oo z or z s y. Th s, w chg h sg of h mgry pr d us srsk or l ovr h compl vrl o do h compl coug. A sysm of dfos h s dvlopd for compl vrls c rdly ppld o rl vrls y sg h compl pr qul o zro. Furhrmor, h compl coug of rl umr s us h umr. If w s y = h dfo, z = z = y, w g z y= =. A vcor h cosss of compl compos, z, wll hv compl coug, z, whos compos r h compl cougs of h compos of h orgl vcor, z. Th sc dfo of lr vcor spc smply ss h vcors mus ssfy cr smpl proprs. Ths r lsd low.. If d y r vcors h spc h + y s lso vcor h spc. Somms h symol s usd o mphsz h h ddo opro for vcors my dffr from h usul opro w pc of ddg ch compo h vcor,, o h corrspodg compo of h vcor o g h compos of h vcor sum +. W wll o cosdrg such

33 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg. Th ddo opro h prvous sm s commuv d ssocv. Th s, + y = y + d + y + z = + y + z = + y + z.. Th spc cos ull lm,, such h + = + =.. For ch vcor,, h spc hr s ohr vcor, -, such h + - =. 5. Vcors c mulpld y sclrs. If d y r vcors h spc d d r sclrs, ll h followg rloshps hold:,, y, d y r ll vcors h spc = + = + v = = v y = + y v = 6. Th orm of vcor,, prssd s, s msur of h sz of h vcor. Ths s grlzo of h usul dfo of h lgh of vcor, = possl orms. Ay dfo of orm mus ssfy h followg rloshps: = for compl > f = f = + y + y 7. A commo dfo of h orm hs h form q q. Thr r my. Ths s clld h q orm d s usully wr s q. I hs oo, h usul dfo of vcor lgh s h wo orm,. Ohr commo orms r h o orm, whch s smply h sum of solu vlus d h fy orm whch s h lm whch hs h mmum solu vlu. 8. Th vcor do produc s grlzd for src vcor spcs usg h oo,y. Somms h oo <,y> s usd. Ths r producs ssfy h followg rloshps. y = y + yz = zyz = f d oly f = src opros hr. I smlr ss h symol s usd o rprs grl opro of mulplco whch my dffr from orml mulplco of wo umrs.

34 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg v > ulss = v y = y = [y [y yy 9. W c form lr comos of y umr, k, vcors h spc. Th lr como s dfd rms of s of k sclrs,,,, k, such h our lr como s gv y h k quo. A s of vcors s sd o lrly dpd f, whr ls o of h s o qul o zro.. A lrly dpd s of vcors s o h s o lrly dpd.. Th vcor spc s sd o -dmsol f s of lrly dpd vcors ss h spc, u o s of + lrly dpd vcors ss h spc.. Ay vcor -dmsol spc c rprsd y lrly dpd como of vcors. Such s of vcors s clld ss s d s sd o sp h spc.. Two vcors whos r produc quls zro r sd o orhogol. Th s, d y r orhogol f, y =. 5. A s of vcors,,,,, r sd o orhogol f h r produc of y ulk pr of vcors vshs. Th s f, = for y d such h, h s of vcors s orhogol. 6. A s of vcors,,,,, r sd o orhoorml f h r produc of y ulk pr of vcors vshs d h r produc of lk vcors quls o. For orhoorml s, h,, =. 7. W c covr y orhogol s of vcors o orhoorml s y dvdg ch compo of h orhogol vcor wh d, y h r produc for h vcor wh slf,,. Th sms ov summrz svrl dfos ou src vcor spcs, orms d r producs. A mor d h s usd hroughou grg lyss courss s h d h w c rprs vcor rms of ss s, whch s lrly dpd s of vcors h sp h spc. Ths ms h y vcor h spc c prssd s lr como of h ss s. W kow h hs s ru for our covol vcors whr w us h ss s,, d k. Th possl do producs of hs ss s c s o form orhoorml s. As od rlr, hs possl do producs r = = k k = d = k = = k = k = k =. I mor grl oo w c rprs hs ss s s =,, =, d = k. Ths ss s s orhoorml sc, =. W wll d h oo of r producs d orhogoly o fucos. For fucos, h r produc s dfd rms of grl. If w hv s of fucos, f dfd o rvl, w df h r produc for hs fucos s follows. k, f f f f d []

35 Vcors d mrcs L. S. Cro, Augus 8, 7 Pg 5 I hs dfo, h fuco s clld h wghg fuco. I my css, = d s o cosdrd h dfo of h r produc of fucos. Th fucos f r orhogol f h followg rloshp holds. d f f f f, [] W c df s of orhoorml fucos y h followg quo. d f f f f, [] Ay s of orhogol fucos g c covrd o s of orhoorml fucos f y dvdg y h squr roo of h r produc, g,g. d g g g g g g f, [] W c show h h fucos f form orhoorml s y cosrucg h r produc f, f.,,,,,, g g g g g g g g g g f f [5] No h h r produc g,g s cos so h w c rg ousd h r produc clculo for f,f. W s h h r produc of f wh slf s qul o o. Th r produc f, f s proporol o g, g whch s zro. Thus h fucos f form orhoorml s.