DERIVATIVES OF KRONECKER PRODUCTS THEMSELVES BASED ON KRONECKER PRODUCT AND MATRIX CALCULUS

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1 Jourl of heoretcl d ppled Iformto echology th Februry 3. Vol. 48 No. 5-3 JI & S. ll rghts reserved. ISSN: E-ISSN: DERIVIVES OF KRONECKER PRODUCS HEMSEVES SED ON KRONECKER PRODUC ND MRIX CCUUS XIOFENG WNG WENYN YNG OGUNG SUN College of Mthemtcl d Physcl Sceces Chogg Uversty of Scece d echology Chogg Ch SRC I some felds Kroecker product hs bee used extesvely. I ths pper we frst revew propertes d deftos of Kroecker. he we derve two propertes of the dervtves of mtrces wth respect to mtrces terms of the proposed cocept. Flly ths pper presets ovel method d ew sght for provg dervtves of Kroecker product themselves by usg the cocepts d propertes of Kroecker Products d Mtrx Clculus. Keywords: Kroecker product; Mtrx Clculus; Vec opertor; Dervtves of Kroecker products. INRODUCION he terest the Kroecker product hs grow recetly. Kroecker product whch ws med fter Germ mthemtc eopold s specl opertor for multplcto of two mtrces. It s mportt for us tht Kroecker product smplfes the otto of my lgorthms. Kroecker product rses my dfferet res of scece d egeerg[4] whch llows more elegt d compct dervtos d hs mportt pplctos my felds cludg computer vso[6] sttstcs[9] cotrol d mtrx euto[]. Especlly the use of my formto theores for Kroecker product s used wdely. Severl treds the developmet of scetfc computg suggest tht Kroecker product opertor wll hve greter role to ply the future. ut the rules d propertes of Kroecker product re lttle dscussed eve books o mthemtcl spects lttle dscussg the propertes d pplctos of Kroecker product very short wthout y expltos of ts rules d propertes. ht s why we re dscussg the rules d propertes. I ths pper we troduce some cocepts of Kroecker product d ts pplcto expressg smplfyg d mplemetg Mtrx Clculus. We dscuss some results whch wll be foud very useful for the developmet of the theory of both Kroecker product d mtrx dfferetto. Flly we prove dervtves of Kroecker product themselves. he pper s orgzed s follows: I Secto we revew brefly some propertes of Kroecker product d the vec opertor whch together provde compct otto. Secto 3 the derve two propertes of the dervtves of mtrces wth respect to mtrces. I Secto 4 we cheve dervtves of Kroecker product themselves. Secto 5 cocludes.. DEFINIIONS ND PROPERIES OF KRONECKER PRODUC et us revew some bsc cocepts of Kroecker product d mtrx clculus for uderstdg the proofs of the theorems preseted the followg sectos whch re useful to estblsh dervtves of Kroecker products themselves []. Defto.. et R R the Kroecker product of d s defed s the mtrx[3] Kroecker product s lso kow s drect product or tesor product. m R m m () 378

2 Jourl of heoretcl d ppled Iformto echology th Februry 3. Vol. 48 No. 5-3 JI & S. ll rghts reserved. ISSN: E-ISSN: R Defto.. et the the opertor vec ( s defed to crete vector by strgg together oe-by-oe the colums of mtrx R [5] (:) () (: ) heorem.. For y three mtrces p R d X R X) ( X ) heorem.. Defe the mtrx R (3) tht trsforms vec ( to vec ( ) : m s the mtrx ) (4) m 3. DERIVIVES OF KRONECKER PRO- DUCS heorem 3.. Determe the dervtve of where s m ( I ) m (5) Proof. Usg the reltoshp betwee the vec d Kroecker product opertors ths c be used to determe the dervtve of where s m. ( I ) ( I) ) (6) ht we defe dervtves of mtrces wth respect to mtrces s cheved by vectorzg the mtrces herefore dervtves of mtrces ( X) dx re the sme s dvec ( ( X )) d X ) the we use the product rule d get d ( I ) m d( (7) hs c be smplfed hus I ) ( I ) (8) ( m ( I + m )( I ) (9) heorem 3.. Determe the dervtve of where s m ( I ) () 4. DERIVIVES OF KRONECKER PRO- DUCS HEMSEVES I some felds t s usully useful to estblsh dervtves of Kroecker products themselves [8]. herefore we cheve dervtves of Kroecker products themselves by usg Kroecker Products d Mtrx Clculus. I ths secto we derve formuls bout Kroecker products themselves s follow: heorem 4.. For y two mtrces R defed s follow: where the mtrx product I R s 379

3 Jourl of heoretcl d ppled Iformto echology th Februry 3. Vol. 48 No. 5-3 JI & S. ll rghts reserved. ISSN: E-ISSN: p p R I p R m Proof. et he wth the defto of Kroecker product we vectorze the mtrx. ht s ) [ p m m p m p m p ] d lkewse [ m m ] m ecuse ( X ) dx s the sme thg s dvec ( ( X )) d X ). hs s where the reltoshp betwee the vec opertor d Kroecker products s useful []. ht s d( )) d( ) () Wth the propertes of the Kroecker product we derve dfferet expresso. he dervtve of ) s d( )) d( ) mp he we hve m d( )) d( ) p m m m m m m m m p m mp m () 38

4 Jourl of heoretcl d ppled Iformto echology th Februry 3. Vol. 48 No. 5-3 JI & S. ll rghts reserved. ISSN: E-ISSN: hus t c be wrtte compctly s where p p 5. CONCUSION I I p m he Kroecker producer whch s very powerful mtrx multplcto tool hs becme more d more mportt pro-blems vrous felds such s 3D Computer Vso pro-blems sttstcs ecoomcs cotrol mtrx euto d so o. I ths pper we hve preseted ovel pproch for curg dervtves of Kroecker products themselves. We strt by troducg the Kroecker product d descrbg the cocepts d propertes of the Kroecker product d coclude by descrbg oe pplcto of the Kroecker product. the we derve two propertes of the dervtves of mtrces wth respect to mtrces d flly compute dervtves of Kroecker products themselves. hs work provdes ew sghts to curg dervtves of Kroecker products themselves. CKNOWEDGEMENS hs work s supported by the Scece d echology Foudto of the Educto Deprtmet of Chogg (KJ44) Iovto tem of Chogg Uversty of Scece & echology (893) d Reserch Foudto of Chogg Uversty of Scece & echology (CKZ5). REFERENCES: [] H. C. Rews J. Ke Kroecker mtrces computer mplemetto d geerlzed spectr J. ssoc. Comput. Mch. Vol.7 97 pp [] Chrles F.V o. Compulttol frmeworks for the Fst Fourer rsform. SIM 99. [3] bdr K.M Mgus J.R. Mtrx lgebr Cmbrdge Uversty Press 5. [4] G.Strg Computtol Scece d Egeerg Wellesley Cmbrdge Press 7. [5] P. cster M. smeetsky he heory of Mtrces secod ed. cdemc Press New York 985. [6] dre Fusello mtter of otto: Severl uses of the Kroecker product 3D computer vso Moleculr Evoluto d Phylogeetcs Ptter Recogto etters Vol.8 7 pp [7] Fju Yu ew method to costruct the tegrble couplg system for dscrete solto euto wth the Kroecker product Physcs etters Vol.37 8 pp [8] Nguye V Khg Kroecker product d ew mtrx form of grg eutos wth multplers for costred multbody systems 38

5 Jourl of heoretcl d ppled Iformto echology th Februry 3. Vol. 48 No. 5-3 JI & S. ll rghts reserved. ISSN: E-ISSN: Mechcs Reserch Commuctos Vol.38 pp [9] my N.gvlle Wllm J.Stewrt he Kroecker product d stochstc utomt etworks Jourl of Computtol d ppled Mthemtcs Vol.67 4 pp [] Dold W. Fusett Chrles. Fulto Hy Hshsh Improved prllel QR method for lrge lest sures problems volvg Kroecker roducts Jourl of Computtol d ppled Mthemtcs Vol pp [] dre Fusello mtter of otto: Severl uses of the Kroecker product 3D computer vso Ptter Recogto etters Vol.8 7 pp [] J. Ngy M. Ng. Perroe Kroecker product pproxmtos for mge restorto wth reflexve boudry codtos SIM J. Mtrx l. ppl. Vol. 6 4 pp