actuary )uj, problem L It X Projection approximation Orthogonal subspace of !- ( iii. ( run projection of Pcxzll rim orthogonal ( Nuu, y H, ye W )
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1 dmensonal ( Note thot rm t w ; n ( ( run u! ( t E 21 k f } actuary uj n ( Nuu h rate h Hutt O Snce when Truth Klment we also have wht Lu te It k} Orthogonal Projecton let W be an un subspace a Eucldean Space V X V o a ar PG Far PG W projecton space orthogonal W on Pe an element n W whch solves best appromaton problem mn L It X y H ye W It Pczll
2 Then In In order to construct P we f a bas ( Ve Nm on c W so that PG I a V We now want that r P C I W that on tf k E LL 3 n C Nw Z d N > O r Th a lnear n m Unknowns ( da 22 system m equaton ( one for each k dm matr form th Gram matr Gly Nm and t depends on bas n W Observe Hud Gram matr wth respect to an orthonormal bas C wa We Wm c W agrees wth G ( we Iwm Im so that n PG Z I w W The constructon P gves that Prep ( projecton Let V be a vector space and Well a sub vector space V tt V F! n E W y TW X qty ( In feet PG t ( Pc
3 How to represent P 112 o W If we f censured bas n R and ( Wr Iwm an orthonormal bas n W we have PM C w w Eg wt w ( IE wt w 9 P ' w wt e RT matr whch represents orthogonal projecton we r to standard bas n R and bas ( we Wm n W Eample We want to fnd matr P orthogonal projecton E 3 X representng IR ' on plane 0 wth respect to come Note wnd bas that vector v / I E and thot a bas for E gven by 4 ( E fg We want to make use 2 system ( v Nz orthogonal out
4 no Orthogonal Zohar w Em E I! er r H u Erm IE Ipso n f! HE; we ET 4 fg whch gves P 's (1 ( 230 t Us b L t H I
5 w WH w Pe For Best appromaton Proposton Let V be a vector space XE V let PG be projecton on a sub vector space WCV Then t holds two W Wnt PG H lls It WH In or words PCA best appromaton among vectors n W Pro By Pythagoras let v Untv HI Kull 't HV 112 By proposton ( proj we have Xy PG EW y PG L W Then N PG 11! Ily 112 and A we W A ( Pc ty Pc 't Hy 1123 Ily O wth equalty only for w PG
6 AAI Orthogonal transformatons We dcuss here ory those lner maps T V o V whch preserve norm vectors e H Tcu H It v 11 tve V We consder only case V HE endowed wth usual Eucldean scalar product ( env utv and norm Hull F Let Te AX The followng holds true Proposton The followng ( T an ometry statements are equvalent t ER HAXH ( T leaves scaler product nvarant t ye IR ( A Ay ( y C A an orthogonal matr ATA In Cv The rows A form an orthonormal bas ( v The columns A form an orthonormal bas 4 We do not prove th proposton
7 out Eample The matrces A ( Tna II Aef?% A are orthogonal matrces n C check t t descrbes counterclockwe rotaton An descrbes reflecton wth respect straght lne sn (E q to cf geo ( check t! Note thot out f An I out f Az I Of course we already observed thot for matrces At A In we have det (A orthogonal f At out CID Colet CA! I D dt AE f 113 The set Och LA ER ATA In endowed wth usual multplcaton between matrces a group t called orthogonal group The set SOG C OCD defund by SOC n L AE Och out A I } a subgroup O ( n and t called Specal Orthogonal Group A matr Ae SO Ch descrbes a rotaton n IR
8 out At AAT AT A At Consder as an eample a matr A E So (3 Such a matr descrbes a rotaton vectors n IR ' around a rotaton a through orgn To fnd such an a we need to fnd set those vectors V whch reman some under such a transformaton e AV V ( t How to solve Hus problem? (2 Is V o P out ( A Iz out f A dt ( A ( I out A C Iz out ( Is A u ~ t ( Iz T ( Iz D det ( A o Iz From (1 we have ( A c n Polt f A Is As a result system has a soluton V 't 0 descrbes drecton rotaton a
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