Iteredite Applictio of Vector d Mtrice Ed Stek Itroductio We decribe iteredite opertio d pplictio of vector d trice for ue i ttitic The itroductio i iteded for thoe who re filir with bic trix lgebr Followig thi itroductio, we illutrte how vector d trice c be ued i iple ttiticl pplictio We decribe i thi docuet the ter: Kroecker product of trice Direct u of trice vec of trix rk of trix igulr d o-igulr trice idepotet trice trce of trix qudrtic for prtitioed trice the opertio: ivere of o-igulr trix ( x trix, digol trix, pttered trix, prtitioed trix) derivtive of product of vector d trice The Kroecker Product (or direct or teor product) of Two trice: The Kroecker product of the trice, A ( ij ) d ( bij ) trix-itereddoc //006 7:0 AM p q B i fored by ultiplyig ech eleet of A by the trix B, d repreeted by the Kroecker product B B L B Thu, B B L B A B Siilrly, p q M M O M B B L B ( B C) ( B C) L ( B C) p q r p q r p q r ( B C) ( B C) L ( B C) p q r p q r p q r A B C p q r M M O M ( B C) ( B C) L ( B C) p q r p q r p q r
0 0 0 0 0 0 Exple : Let A I d B The A B 0 0 0 0 0 0 The Direct Su of Mtrice: The direct u of trice i equl to block digol trix with the idividul trice the digol block For the trice A, A d A, Ai i 0 A 0 0 0 A b A 0 0 where the trice 0 cofor to the row d colu dieio of the other trice c d e f Exple: Suppoe A, A,, d A The The Vec of Mtrix 0 L 0 0 L 0 M M O M 0 0 L Ai i The vec of tix crete colu vector fored by tckig the colu i the trix, A L where ech loctig oe colu uder other For exple, uppoe trix-itereddoc //006 7:0 AM vector, j i of dieio The vec ( A ) will hve dieio A ueful M reult for the vec of product of trice i the followig: vec ABC C A vec B Lier Spce, Sp, Rk, Bi, Dieio, Sigulr, No-igulr: A L if of the for The colu pce of trix x+ x + + x
Row pce re defied i iilr er for row A lier pce i o-epty et, V, of ll trice of the e dieio uch tht the u of y two trice i the pce, d the product of clr tie y trix i i the pce The p of et of trice i the et of ll trice tht c be fored by lier cobitio of the trice The bi of lier pce i the et of lierly idepedet trice tht p the pce Ay two be for the e lier pce coti the e uber of trice The uber of trice i bi i the dieio of the lier pce The row rk of trix i the dieio of the row pce of the trix The colu rk of trix i the dieio of the colu pce of the trix rk ( A ) A trix A i id to hve full colu rk (be o-igulr) if ( A ) <, the the trix A i igulr Idepotet Mtrice: A qure trix i idepotet if AA A rk If Exple: Let A I J The AA I JI J I J J + J I J A Trce of trix: The trce of qure trix i the u of the digol eleet of the trix Exple: Let ( ij ) A The i trce A tr A ii The trce of product of trice i give by: tr tr ( ) tr tr ( ) Alo tr tr tr ABC CAB BCA AB BA BA AB Qudrtic For: A qudrtic for i clr give by x A x Expected vlue of qudrtic for Let A be o-igulr yetric trix, d let Y be E vr Y Σ We vector of rdo vrible where ( Y ) µ d evlute E ( YAY ) Firt, ote tht tr ( ) tr ( ) vlue iide the trce opertor A reult, E( ) tr E( ) trix-itereddoc //006 7:0 AM YAY YY A We c brig the expected YAY YY A Now, recll tht by
YY Y + Y Y Σ + µµ defiitio, vr ( Y) E( YY ) E( Y) E( Y ) Thu, E vr E E Replcig thi expreio bove, E( ) tr( + ) tr + tr( ) YAY ΣA µµ A ΣA µµ A Filly, YAY ΣA + µ Aµ re-rrgig the lt ter i the trce, E tr Prtitioed Mtrix: A prtitioed trix i trix repreeted by ub-tric i rry 0 0 0 0 0 0 A A Exple: Let A 0 0 0 The the trix A c be prtitioed ito 0 0 0 A A 0 0 0 0 the ub-trice give by A for i, j,, ij Ivere of qure o-igulr Mtrix: The ivere of qure o-igulr trix i repreeted by XX I X uch tht X X Id Exple : Digol Mtrix Let 0 0 C 0 4 0 The 0 0 5 0 0 4 0 0 5 C 0 0 Exple : Two by Two trix: Let b A The c d A d b d bc c Exple : Su of two trice: Let R be qure o-igulr trix, d colu vector The R+ u R + ur R ur d u be Specil Ce : I J I + J N N / Specil Ce : Suppoe tht ZG d u The u ZGZ d R+ u R+ ZGZ trix-itereddoc //006 7:0 AM 4
( ) R+ u R + ur R ur The ( + ) ( + ) / / R ZGZ R G Z R ZG R ZGZ R Exple 4: Ivere of o-igulr prtitioed trix: Let X A + A BQ CA A BQ where Q CA Q The Ivere of A Product of Mtrice: Q D CA B A B X C D The The ivere of product of qure, o-igulr trice i fored by the product of thef ABC C B A the ivere of the idividul trice i revere order: Derivtive: Differetitio with repect to vector i iilr to differetitio i geerl Derivtive of clr by vector: Let d x be two vector The x i clr We defie x x x x x x M x x Derivtive of trix by vector: Ab We defie the derivtive of A b Exple: Let b A, b The b trix-itereddoc //006 7:0 AM 5
jbj j b b Ab jbj b b We defie j b jbj j b b b Ab b b b b b b b b b b A Derivtive of qudrtic for: b Let b d Q (( q ij )), uch tht qudrtic for i give by bqb The b bqb Qb b trix-itereddoc //006 7:0 AM 6