Dobutok Hatri Rao angl Khatri Rao product matrichna operaciya peremnozhennya matric sho viznachayetsya virazom 1 2 A B A i j B i j i j displaystyle mathbf A ast mathbf B left mathbf A ij otimes mathbf B ij right ij v yakomu ij j blok yavlyaye soboyu dobutok Kronekera mipi njqj vidpovidnih blokiv A i B za umovi sho kilkist ryadkiv i stovpciv oboh matric odnakova Rozmirnist dobutku Si mipi Sj njqj Napriklad yaksho matrici A i B mayut blokovu rozmirnist 2 2 A A 11 A 12 A 21 A 22 1 2 3 4 5 6 7 8 9 B B 11 B 12 B 21 B 22 1 4 7 2 5 8 3 6 9 displaystyle mathbf A left begin array c c mathbf A 11 amp mathbf A 12 hline mathbf A 21 amp mathbf A 22 end array right left begin array c c c 1 amp 2 amp 3 4 amp 5 amp 6 hline 7 amp 8 amp 9 end array right quad mathbf B left begin array c c mathbf B 11 amp mathbf B 12 hline mathbf B 21 amp mathbf B 22 end array right left begin array c c c 1 amp 4 amp 7 hline 2 amp 5 amp 8 3 amp 6 amp 9 end array right otrimayemo A B A 11 B 11 A 12 B 12 A 21 B 21 A 22 B 22 1 2 12 21 4 5 24 42 14 16 45 72 21 24 54 81 displaystyle mathbf A ast mathbf B left begin array c c mathbf A 11 otimes mathbf B 11 amp mathbf A 12 otimes mathbf B 12 hline mathbf A 21 otimes mathbf B 21 amp mathbf A 22 otimes mathbf B 22 end array right left begin array c c c c 1 amp 2 amp 12 amp 21 4 amp 5 amp 24 amp 42 hline 14 amp 16 amp 45 amp 72 21 amp 24 amp 54 amp 81 end array right Zmist 1 Stovpcevij dobutok Hatri Rao 1 1 Zastosuvannya 2 Torcevij dobutok 3 Osnovni vlastivosti 4 Blokovij torcevij dobutok 4 1 Osnovni vlastivosti 4 2 Zastosuvannya 5 Div takozh 6 Primitki 7 DzherelaStovpcevij dobutok Hatri Rao red Stovpcevij dobutok Kronekera dvoh matric takozh prijnyato nazivati dobutkom Hatri Rao Cej dobutok peredbachaye sho bloki matric ye yih stovpcyami V takomu vipadku m1 m p1 p n q i dlya kozhnogo j nj pj 1 Rezultatom dobutku ye mp n matrica kozhen stovpec yakoyi otrimuyetsya yak dobutok Kronekera vidpovidnih stovpciv matric A i B Spirayuchis na rozbittya matric z poperednogo prikladu na stovpci otrimayemo C C 1 C 2 C 3 1 2 3 4 5 6 7 8 9 D D 1 D 2 D 3 1 4 7 2 5 8 3 6 9 displaystyle mathbf C left begin array c c c mathbf C 1 amp mathbf C 2 amp mathbf C 3 end array right left begin array c c c 1 amp 2 amp 3 4 amp 5 amp 6 7 amp 8 amp 9 end array right quad mathbf D left begin array c c c mathbf D 1 amp mathbf D 2 amp mathbf D 3 end array right left begin array c c c 1 amp 4 amp 7 2 amp 5 amp 8 3 amp 6 amp 9 end array right nbsp i dali C D C 1 D 1 C 2 D 2 C 3 D 3 1 8 21 2 10 24 3 12 27 4 20 42 8 25 48 12 30 54 7 32 63 14 40 72 21 48 81 displaystyle mathbf C ast mathbf D left begin array c c c mathbf C 1 otimes mathbf D 1 amp mathbf C 2 otimes mathbf D 2 amp mathbf C 3 otimes mathbf D 3 end array right left begin array c c c 1 amp 8 amp 21 2 amp 10 amp 24 3 amp 12 amp 27 4 amp 20 amp 42 8 amp 25 amp 48 12 amp 30 amp 54 7 amp 32 amp 63 14 amp 40 amp 72 21 amp 48 amp 81 end array right nbsp Zastosuvannya red Stovpceva versiya dobutku Hatri Rao zastosovuyetsya v linijnij algebri dlya analitichnoyi obrobki danih 3 i optimizaciyi rishen problemi obernennya diagonalnih matric 4 5 V 1996 r stovpcevij dobutok Hatri Rao buv zaproponovanij dlya formalizaciyi zadachi ocinyuvannya napryamku prihodu ta chasu zatrimki signaliv v cifrovij antennij reshitci 6 a takozh dlya opisu vidguku 4 koordinatnogo radara 7 Torcevij dobutok red nbsp Torcevij dobutok matricAlternativna koncepciya dobutku matric yaka na vidminu vid stovpcevoyi versiyi dobutku Hatri Rao vikoristovuye rozbittya matric na ryadki bula zaproponovana Slyusarem V I 8 v 1996 r i nazvana nim torcevij dobutok angl face splitting product 7 9 10 11 abo transponovanij dobutok Hatri Rao angl transposed Khatri Rao product 12 Cej tip matrichnogo dobutku spirayetsya na peremnozhennya elementiv ryadkiv dvoh i bilshe matric z odnakovoyu kilkistyu ryadkiv za pravilom dobutku Kronekera Vikoristovuyuchi rozbittya matric z poperednih prikladiv na ryadki C C 1 C 2 C 3 1 2 3 4 5 6 7 8 9 D D 1 D 2 D 3 1 4 7 2 5 8 3 6 9 displaystyle mathbf C left begin array c c mathbf C 1 hline mathbf C 2 hline mathbf C 3 end array right left begin array c c c 1 amp 2 amp 3 hline 4 amp 5 amp 6 hline 7 amp 8 amp 9 end array right quad mathbf D left begin array c mathbf D 1 hline mathbf D 2 hline mathbf D 3 end array right left begin array c c c 1 amp 4 amp 7 hline 2 amp 5 amp 8 hline 3 amp 6 amp 9 end array right nbsp mozhna zapisati 7 9 10 C D C 1 D 1 C 2 D 2 C 3 D 3 1 4 7 2 8 14 3 12 21 8 20 32 10 25 40 12 30 48 21 42 63 24 48 72 27 54 81 displaystyle mathbf C bullet mathbf D left begin array c mathbf C 1 otimes mathbf D 1 hline mathbf C 2 otimes mathbf D 2 hline mathbf C 3 otimes mathbf D 3 end array right left begin array c c c c c c c c c 1 amp 4 amp 7 amp 2 amp 8 amp 14 amp 3 amp 12 amp 21 hline 8 amp 20 amp 32 amp 10 amp 25 amp 40 amp 12 amp 30 amp 48 hline 21 amp 42 amp 63 amp 24 amp 48 amp 72 amp 27 amp 54 amp 81 end array right nbsp Osnovni vlastivosti red Transponuvannya Slyusar V I 1996 7 9 13 A B T A T B T displaystyle left mathbf A bullet mathbf B right textsf T textbf A textsf T ast mathbf B textsf T nbsp Komutativnist i asociativnist 7 9 13 A B C A B A C B C A B A C A k A B A k B k A B A B C A B C displaystyle begin aligned mathbf A bullet mathbf B mathbf C amp mathbf A bullet mathbf B mathbf A bullet mathbf C mathbf B mathbf C bullet mathbf A amp mathbf B bullet mathbf A mathbf C bullet mathbf A k mathbf A bullet mathbf B amp mathbf A bullet k mathbf B k mathbf A bullet mathbf B mathbf A bullet mathbf B bullet mathbf C amp mathbf A bullet mathbf B bullet mathbf C end aligned nbsp de A B i C matrici a k skalyar a B B a displaystyle a bullet mathbf B mathbf B bullet a nbsp 13 de a displaystyle a nbsp vektor z tiyeyu zh kilkistyu elementiv sho i kilkist ryadkiv matrici B displaystyle mathbf B nbsp Vlastivist zmishanogo dobutku 1997 13 A B A T B T A A T B B T displaystyle mathbf A bullet mathbf B left mathbf A textsf T ast mathbf B textsf T right left mathbf A mathbf A textsf T right circ left mathbf B mathbf B textsf T right nbsp A B C D A C B D displaystyle mathbf A bullet mathbf B mathbf C ast mathbf D mathbf A mathbf C circ mathbf B mathbf D nbsp 10 A B C D L M N P A L B M C N D P displaystyle mathbf A bullet mathbf B bullet mathbf C bullet mathbf D mathbf L ast mathbf M ast mathbf N ast mathbf P mathbf A mathbf L circ mathbf B mathbf M circ mathbf C mathbf N circ mathbf D mathbf P nbsp 12 A B T A B A T A B T B displaystyle mathbf A ast mathbf B textsf T mathbf A ast mathbf B mathbf A textsf T mathbf A circ mathbf B textsf T mathbf B nbsp 14 de displaystyle circ nbsp oznachaye dobutok Adamara A B C D A C B D displaystyle mathbf A circ mathbf B bullet mathbf C circ mathbf D mathbf A bullet mathbf C circ mathbf B bullet mathbf D nbsp 13 A B C A B C displaystyle mathbf A otimes mathbf B bullet mathbf C mathbf A otimes mathbf B bullet mathbf C nbsp 7 A B C D A C B D displaystyle mathbf A otimes mathbf B mathbf C ast mathbf D mathbf A mathbf C ast mathbf B mathbf D nbsp 14 A B C D A C B D displaystyle mathbf A bullet mathbf B mathbf C otimes mathbf D mathbf A mathbf C bullet mathbf B mathbf D nbsp 10 15 A L B M C S A B C L M S displaystyle mathbf A bullet mathbf L mathbf B otimes mathbf M mathbf C otimes mathbf S mathbf A mathbf B mathbf C bullet mathbf L mathbf M mathbf S nbsp c T d T c T d T displaystyle c textsf T bullet d textsf T c textsf T otimes d textsf T nbsp 13 c d c d displaystyle c ast d c otimes d nbsp de c displaystyle c nbsp i d displaystyle d nbsp ye vektorami uzgodzhenoyi rozmirnosti A c T d A d T c displaystyle mathbf A ast c textsf T d mathbf A ast d textsf T c nbsp 16 d T c A T c T d A T displaystyle d textsf T c bullet mathbf A textsf T c textsf T d bullet mathbf A textsf T nbsp A B c d A c B d displaystyle mathbf A bullet mathbf B c otimes d mathbf A c circ mathbf B d nbsp 17 de c displaystyle c nbsp i d displaystyle d nbsp ye vektorami uzgodzhenoyi rozmirnosti A B M N c Q P d A M N c B Q P d displaystyle mathbf A bullet mathbf B mathbf M mathbf N c otimes mathbf Q mathbf P d mathbf A mathbf M mathbf N c circ mathbf B mathbf Q mathbf P d nbsp F C 1 x C 2 y F C 1 F C 2 x y F C 1 x F C 2 y displaystyle mathcal F C 1 x star C 2 y mathcal F C 1 bullet mathcal F C 2 x otimes y mathcal F C 1 x circ mathcal F C 2 y nbsp de displaystyle star nbsp ye simvolom vektornoyi zgortki i F displaystyle mathcal F nbsp matricya diskretnogo peretvorennya Fur ye totozhnist ye rozvitkom vlastivosti vidlikovogo sketcha 18 A B A 1 c T 1 k T B displaystyle mathbf A bullet mathbf B mathbf A otimes mathbf 1 c textsf T circ mathbf 1 k textsf T otimes mathbf B nbsp 19 de A displaystyle mathbf A nbsp r c displaystyle r times c nbsp matricya B displaystyle mathbf B nbsp r k displaystyle r times k nbsp matricya 1 c displaystyle mathbf 1 c nbsp 1 k displaystyle mathbf 1 k nbsp vektori z c displaystyle c nbsp ta k displaystyle k nbsp odinic vidpovidno M M M 1 T 1 T M displaystyle mathbf M bullet mathbf M mathbf M otimes mathbf 1 textsf T circ mathbf 1 textsf T otimes mathbf M nbsp 20 de M displaystyle mathbf M nbsp ye r c displaystyle r times c nbsp matriceyu displaystyle circ nbsp dobutok Adamara i 1 displaystyle mathbf 1 nbsp vektor z c displaystyle c nbsp odinic M M M M 1 T displaystyle mathbf M bullet mathbf M mathbf M circ mathbf M otimes mathbf 1 textsf T nbsp de displaystyle circ nbsp simvol pronikayuchogo torcevogo dobutku matric Analogichno P N P 1 c 1 k N displaystyle mathbf P ast mathbf N mathbf P otimes mathbf 1 c circ mathbf 1 k otimes mathbf N nbsp de P displaystyle mathbf P nbsp c r displaystyle c times r nbsp matricya N displaystyle mathbf N nbsp k r displaystyle k times r nbsp matricya W d A w A displaystyle mathbf W d mathbf A mathbf w bullet mathbf A nbsp 13 v e c w T A B B T A w displaystyle vec mathbf w textsf T ast mathbf A mathbf B mathbf B textsf T ast mathbf A mathbf w nbsp 10 v e c A w B B T A w displaystyle vec mathbf A mathbf w bullet mathbf B mathbf B textsf T ast mathbf A mathbf w nbsp 12 v e c A T W d A A A T w displaystyle vec mathbf A textsf T mathbf W d mathbf A mathbf A bullet mathbf A textsf T mathbf w nbsp 20 v e c A W d A T A T A T T w A A w displaystyle vec mathbf A mathbf W d mathbf A textsf T mathbf A textsf T bullet mathbf A textsf T textsf T mathbf w mathbf A ast mathbf A mathbf w nbsp de w displaystyle mathbf w nbsp vektor utvorenij iz diagonalnih elementiv matrici W d displaystyle mathbf W d nbsp v e c A displaystyle vec mathbf A nbsp operaciya formuvannya vektora z matrici A displaystyle mathbf A nbsp shlyahom roztashuvannya odin pid odnim yiyi stovpciv Vlastivist poglinannya dobutku Kronekera A L B M C S K T A B C K L M S T displaystyle mathbf A bullet mathbf L mathbf B otimes mathbf M mathbf C otimes mathbf S mathbf K ast mathbf T mathbf A mathbf B mathbf C mathbf K circ mathbf L mathbf M mathbf S mathbf T nbsp 10 15 A L B M C S c d A B C c L M S d displaystyle mathbf A bullet mathbf L mathbf B otimes mathbf M mathbf C otimes mathbf S c otimes d mathbf A mathbf B mathbf C c circ mathbf L mathbf M mathbf S d nbsp A L B M C S P c Q d A B C P c L M S Q d displaystyle mathbf A bullet mathbf L mathbf B otimes mathbf M mathbf C otimes mathbf S mathbf P c otimes mathbf Q d mathbf A mathbf B mathbf C mathbf P c circ mathbf L mathbf M mathbf S mathbf Q d nbsp de c displaystyle c nbsp i d displaystyle d nbsp ye vektorami uzgodzhenoyi rozmirnosti Napriklad 17 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 s 1 0 0 s 2 r 1 0 0 r 2 x 1 x 2 y 1 y 2 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 1 s 1 0 0 s 2 x 1 x 2 1 1 1 1 r 1 0 0 r 2 y 1 y 2 1 0 0 1 1 0 1 1 1 1 s 1 0 0 s 2 x 1 x 2 1 0 1 0 0 1 1 1 1 1 r 1 0 0 r 2 y 1 y 2 displaystyle begin aligned amp quad left begin bmatrix 1 amp 0 0 amp 1 1 amp 0 end bmatrix bullet begin bmatrix 1 amp 0 1 amp 0 0 amp 1 end bmatrix right left begin bmatrix 1 amp 1 1 amp 1 end bmatrix otimes begin bmatrix 1 amp 1 1 amp 1 end bmatrix right left begin bmatrix sigma 1 amp 0 0 amp sigma 2 end bmatrix otimes begin bmatrix rho 1 amp 0 0 amp rho 2 end bmatrix right left begin bmatrix x 1 x 2 end bmatrix ast begin bmatrix y 1 y 2 end bmatrix right 5pt amp quad left begin bmatrix 1 amp 0 0 amp 1 1 amp 0 end bmatrix bullet begin bmatrix 1 amp 0 1 amp 0 0 amp 1 end bmatrix right left begin bmatrix 1 amp 1 1 amp 1 end bmatrix begin bmatrix sigma 1 amp 0 0 amp sigma 2 end bmatrix begin bmatrix x 1 x 2 end bmatrix otimes begin bmatrix 1 amp 1 1 amp 1 end bmatrix begin bmatrix rho 1 amp 0 0 amp rho 2 end bmatrix begin bmatrix y 1 y 2 end bmatrix right 5pt amp quad begin bmatrix 1 amp 0 0 amp 1 1 amp 0 end bmatrix begin bmatrix 1 amp 1 1 amp 1 end bmatrix begin bmatrix sigma 1 amp 0 0 amp sigma 2 end bmatrix begin bmatrix x 1 x 2 end bmatrix circ begin bmatrix 1 amp 0 1 amp 0 0 amp 1 end bmatrix begin bmatrix 1 amp 1 1 amp 1 end bmatrix begin bmatrix rho 1 amp 0 0 amp rho 2 end bmatrix begin bmatrix y 1 y 2 end bmatrix end aligned nbsp ta inshi Krim togo Slyusarem V I buli zaproponovani blokovi versiyi transponovanogo dobutku ta doslidzheni yih vlastivosti 7 Blokovij torcevij dobutok red nbsp Zastosuvannya blokovogo transponovanogo torcevogo dobutku dlya opisu vidguku bagatogrannoyi cifrovoyi antennoyi reshitki 15 Dlya blokovih matric z odnakovoyu kilkistyu ryadkiv u vidpovidnih blokah A A 11 A 12 A 21 A 22 B B 11 B 12 B 21 B 22 displaystyle mathbf A left begin array c c mathbf A 11 amp mathbf A 12 hline mathbf A 21 amp mathbf A 22 end array right quad mathbf B left begin array c c mathbf B 11 amp mathbf B 12 hline mathbf B 21 amp mathbf B 22 end array right nbsp zgidno z viznachennyam 7 10 blokovij torcevij dobutok A B displaystyle mathbf A bullet mathbf B nbsp zapishetsya u viglyadi A B A 11 B 11 A 12 B 12 A 21 B 21 A 22 B 22 displaystyle mathbf A bullet mathbf B left begin array c c mathbf A 11 bullet mathbf B 11 amp mathbf A 12 bullet mathbf B 12 hline mathbf A 21 bullet mathbf B 21 amp mathbf A 22 bullet mathbf B 22 end array right nbsp Analogichno dlya blokovogo transponovanogo torcevogo dobutku abo blokovogo stovpcevogo dobutku Hatri Rao dvoh matric A B displaystyle mathbf A ast mathbf B nbsp z odnakovoyu kilkistyu stovpciv u vidpovidnih blokah spravedlivo 7 A B A 11 B 11 A 12 B 12 A 21 B 21 A 22 B 22 displaystyle mathbf A ast mathbf B left begin array c c mathbf A 11 ast mathbf B 11 amp mathbf A 12 ast mathbf B 12 hline mathbf A 21 ast mathbf B 21 amp mathbf A 22 ast mathbf B 22 end array right nbsp Osnovni vlastivosti red Transponuvannya A B T A T B T displaystyle left mathbf A ast mathbf B right textsf T textbf A textsf T bullet mathbf B textsf T nbsp 15 Zastosuvannya red Rodina torcevih dobutkiv matric stala osnovoyu zapochatkovanoyi Slyusarem V I tenzorno matrichnoyi teoriyi cifrovih antennih reshitok dlya radiotehnichnih sistem 12 yaka nadali otrimala rozvitok yak chastina teoriyi cifrovoyi obrobki signaliv Torcevij dobutok nabuv shirokogo poshirennya v sistemah mashinnogo navchannya statistichnij obrobci velikih danih 17 Vin dozvolyaye skorotiti obsyagi obchislen pri realizaciyi metodu zmenshennya rozmirnosti danih sho oderzhav nazvu tenzornij sketch 17 a takozh shvidkogo peretvorennya Dzhonsona Lindenshtrausa 17 Pri comu zdijsnyuyetsya perehid vid matrici velikoyi rozmirnosti do dobutku Adamara sho operuye matricyami menshogo rozmiru Pohibki aproksimaciyi dannih velikoyi rozmirnosti na osnovi torcevogo dobutku matric zadovolnyayut lemi Dzhonsona Lindenshtrausa 17 21 U tomu zh konteksti ideya torcevogo dobutku mozhe buti vikoristana dlya virishennya zavdannya diferencijnoyi privatnosti angl differential privacy 16 Krim togo analogichni obchislennya buli zastosovani dlya formuvannya tenzoriv spivpadan v zadachah obrobki prirodnoyi movi i pobudovi gipergrafiv podibnosti zobrazhen 22 Torcevij dobutok vikoristanij u 2003 r dlya P splajn aproksimaciyi 19 u 2006 r dlya pobudovi uzagalnenih linijnih modelej masiviv danih GLAM pri yih statistichnij obrobci 20 a takozh dlya efektivnoyi realizaciyi yadrovih metodiv mashinnogo navchannya ta doslidzhennya vzayemodiyi genotipiv z otochuyuchim seredovishem 23 Div takozh red Dobutok Kronekera Tenzornij sketch Lema Dzhonsona LindenshtrausaPrimitki red Khatri C G C R Rao 1968 Solutions to some functional equations and their applications to characterization of probability distributions Sankhya 30 167 180 Arhiv originalu za 23 zhovtnya 2010 Procitovano 21 serpnya 2008 Zhang X Yang Z Cao C 2002 Inequalities involving Khatri Rao products of positive semi definite matrices Applied Mathematics E notes 2 117 124 See e g H D Macedo and J N Oliveira A linear algebra approach to OLAP Formal Aspects of Computing 27 2 283 307 2015 Lev Ari Hanoch 1 sichnya 2005 Efficient Solution of Linear Matrix Equations with Application to Multistatic Antenna Array Processing Communications in Information amp Systems EN 05 1 123 130 ISSN 1526 7555 doi 10 4310 CIS 2005 v5 n1 a5 Arhiv originalu za 12 lipnya 2020 Procitovano 12 lipnya 2020 Masiero B Nascimento V H 1 travnya 2017 Revisiting the Kronecker Array Transform IEEE Signal Processing Letters 24 5 525 529 Bibcode 2017ISPL 24 525M ISSN 1070 9908 doi 10 1109 LSP 2017 2674969 Arhiv originalu za 12 lipnya 2020 Procitovano 12 lipnya 2020 Vanderveen M C Ng B C Papadias C B amp Paulraj A n d Joint angle and delay estimation JADE for signals in multipath environments Conference Record of The Thirtieth Asilomar Conference on Signals Systems and Computers DOI 10 1109 acssc 1996 599145 a b v g d e zh i k Slyusar V I 27 grudnya 1996 End products in matrices in radar applications Radioelectronics and Communications Systems 1998 Vol 41 Number 3 50 53 Arhiv originalu za 27 lipnya 2020 Procitovano 27 lipnya 2020 Anna Esteve Eva Boj amp Josep Fortiana 2009 Interaction Terms in Distance Based Regression Communications in Statistics Theory and Methods 38 19 P 3501 1 Arhivovano 26 kvitnya 2021 u Wayback Machine a b v g Slyusar V I 20 travnya 1997 Analytical model of the digital antenna array on a basis of face 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and data mining Association for Computing Machinery doi 10 1145 2487575 2487591 a b Eilers Paul H C Marx Brian D 2003 Multivariate calibration with temperature interaction using two dimensional penalized signal regression Chemometrics and Intelligent Laboratory Systems 66 2 159 174 doi 10 1016 S0169 7439 03 00029 7 a b v Currie I D Durban M Eilers P H C 2006 Generalized linear array models with applications to multidimensional smoothing Journal of the Royal Statistical Society 68 2 259 280 doi 10 1111 j 1467 9868 2006 00543 x Ahle Thomas Kapralov Michael Knudsen Jakob Pagh Rasmus Velingker Ameya Woodruff David Zandieh Amir 2020 Oblivious Sketching of High Degree Polynomial Kernels ACM SIAM Symposium on Discrete Algorithms Association for Computing Machinery doi 10 1137 1 9781611975994 9 Bryan Bischof Higher order co occurrence tensors for hypergraphs via face splitting Published 15 February 2020 Mathematics Computer Science ArXiv Arhivovano 25 listopada 2020 u Wayback Machine Johannes W R Martini Jose Crossa Fernando H Toledo Jaime Cuevas On Hadamard and Kronecker products in covariance structures for genotype x environment interaction Plant Genome 2020 13 e20033 Page 5 2 Dzherela red Khatri C G C R Rao 1968 Solutions to some functional equations and their applications to characterization of probability distributions Sankhya 30 167 180 Arhiv originalu za 23 zhovtnya 2010 Procitovano 21 serpnya 2008 Zhang X Yang Z Cao C 2002 Inequalities involving Khatri Rao products of positive semi definite matrices Applied Mathematics E notes 2 117 124 Matrix Algebra amp Its Applications to Statistics amp Econometrics C R Rao with M Bhaskara Rao World Scientific 1998 P 216 Otrimano z https uk wikipedia org w index php title Dobutok Hatri Rao amp oldid 36821414 Torcevij dobutok
