OpnMx Mtris n Oprtors Sr Mln Mtris: t uilin loks Mny typs? Dnots r lmnt mxmtrix( typ= Zro", nrow=, nol=, nm="" ) mxmtrix( typ= Unit", nrow=, nol=, nm="" ) mxmtrix( typ= Int", nrow=, nol=, nm="" ) mxmtrix( typ= Di", nrow=, nol=, mxmtrix( typ= Si", nrow=, nol=, mxmtrix( typ= Stn", nrow=, nol=, mxmtrix( typ= Symm", nrow=, nol=, mxmtrix( typ= Lowr", nrow=, nol=, mxmtrix( typ= Full", nrow=, nol=,
//00 Mtrix oprtions lssi vx OpnMx Nm Mtmtil Clssi Mx OpnMx Conormility Invrs - ~ solv() r= Trnspos t() Non Powr ^ ^ ^ or %^% Non Str * * %*% Dot.. * r Kronkr @ %x% non Qurti & & %&% Division % % / ition + + + Sutrtion - - - sion _ or _ or rin or in r r r r r r & r r & r & r & r or Invrs mxlr( xprssion=solv(mtrix_nm), nm= "), Only squr mtris my invrt, ut ty my itr symmtri or non-symmtri. T invrs o mtrix is usully writtn - n implis tt - = - = I wr I is t intity mtrix. I t invrs os not xist (possily u to rounin rrors), OpnMx will trmint wit n rror mss. Som prutions n tkn to voi tis, su s supplyin strtin vlus tt llow invrsion, or puttin ounry onstrints on prmtrs to prvnt tir tkin vlus tt woul l to sinulr mtrix. solv
//00 Trnspos mxlr( xprssion=t(mtrix_nm), nm= "), ny mtrix my trnspos. T trnspos o is writtn. T orr o t mtrix ns rom r to r, s t rows om t olumns n vi-vrs. t t Powr mxlr( xprssion=(^), nm=""), ll t lmnts o mtrix my ris to powr usin t ^ symol. Wn mtrix is ris to t powr o mtrix, tis oprtor works t sm wy s t Dot prout (s low), ut lmnts o t irst mtrix r ris to t powr o tos in t son mtrix inst o multipli y tm. ^ ^
//00 Powr mxlr( xprssion= ( %^% ), nm=""), Essntilly, tis oprtor works t sm wy s t Kronkr prout (s low), ut lmnts o t irst mtrix r ris to t powr o tos in t son mtrix inst o multipli y tm. It is possil to us ntiv powrs n non-intr xponnts to init riprol untions n roots o lmnts, ut it is not possil to ris ntiv numr to non-intr powr. %^% Multiplition mxlr( xprssion=( %*% ), nm="c"), %*% or Str is t orinry orm o mtrix multiplition (usully writtn s *). T lmnts o (m n) n (n p) r omin to orm t lmnts o mtrix C(m p) usin t ormul Mtris multipli in tis wy must onorml or multiplition. Tis mns tt t numr o olumns in t irst mtrix must qul t numr o rows in t son mtrix. For xmpl, t mtrix prout %*% % %
//00 Dot prout mxlr( xprssion=( * ), nm="c"), Dot is notr typ o mtrix multiplition, wi is on lmnt y lmnt (usully writtn s. ). For two mtris to multipli in tis wy, ty must v t sm imnsions. Elmnts o t ot prout r sri y t ormul Cij = ij Dij. For xmpl, t ot prout *D is * Kronkr prout mxlr( xprssion=( %x% ), nm="c"), T rit Kronkr prout o two mtris is orm y multiplyin lmnt o y t mtrix. I is o orr (m n) n is o orr (p q), tn t rsult will o orr mp nq. Tr r no onormility ritri or tis typ o prout. For xmpl %x% is 5
//00 Qurti prout mxlr( xprssion=( %&% ), nm="c"), Mny struturl qution n otr sttistil mols us qurti prouts o t orm, n t qurti oprtor (%&% ) is ot simpl n iint wy to implmnt qurtis. Not tt E n ny sp, ut to onorml or qurti prout t mtrix must squr n v t sm numr o olumns s t mtrix E. For xmpl, t qurti prout E%&% % &% %*% %*% i i i j i j Elmnt ivision mxlr( xprssion=( / ), nm="c"), / os lmnt y lmnt ivision. For two mtris to ivi in tis wy, ty must v t sm imnsions. Elmnts o t rsult, C r sri y t ormul For xmpl, t ivision /D is / 6
//00 ition mxlr( xprssion=( + ), nm="c"), ition o mtris is prorm lmnt y lmnt. For two mtris to, ty must v t sm imnsions. Elmnts o t sum, C r sri y t ormul For xmpl Sutrtion mxlr( xprssion=( - ), nm="c"), Sutrtion o mtris is prorm lmnt y lmnt. For on mtrix to sutrt rom notr, ty must v t sm imnsions. Elmnts o t irn, C r sri y t ormul For xmpl 7
//00 Horizontl sion mxlr( xprssion=in (, ), nm="c"), in() llows prtitionin o mtris. Its oprtion is ll orizontl sion us in(,d) is orm y stikin D onto t rit n si o. For two mtris to r in tis wy, ty v to v t sm numr o rows. I (m n) n D (m p) r r, t rsult C is o orr (m (n+p)). For xmpl in 7 5, 8 6 9 5 6 7 8 9 Vrtil sion mxlr( xprssion=rin (, ), nm="c"), rin() llows prtitionin o mtris. Its oprtion is ll vrtil sion us rin(,d) is orm y stikin D unrnt. For two mtris to r in tis wy, ty must v t sm numr o olumns. I (m n) n D (p n) r r, t rsult C is o orr ((m+p) n). For xmpl rin 5, 5 6 7 8 6 7 8 8