Dobutok Kronekera binarna operaciya nad matricyami dovilnogo rozmiru poznachayetsya displaystyle otimes Rezultatom ye blochna matricya Dobutok Kronekera ne slid putati zi zvichajnim mnozhennyam matric Operaciya nazvana na chest nimeckogo matematika Leopolda Kronekera Zmist 1 Viznachennya 2 Bilinijnist asociativnist ta nekomutativnist 3 Transponuvannya 4 Mishanij dobutok 5 Suma ta eksponenta Kronekera 6 Spektr slid ta viznachnik 7 Singulyarnij rozklad ta rang 8 Blokovi versiyi dobutku Kronekera 8 1 Dobutok Trejsi Singha 8 2 Dobutok Hatri Rao 8 3 Torcevij dobutok 9 Div takozh 10 Dzherela 11 PrimitkiViznachennyared Yaksho A matricya rozmiru m n B matricya rozmiru p q todi dobutkom Kronekera ye blochna matricya rozmiru mp nq A B a 11 B a 1 n B a m 1 B a m n B displaystyle A otimes B begin bmatrix a 11 B amp cdots amp a 1n B vdots amp ddots amp vdots a m1 B amp cdots amp a mn B end bmatrix nbsp Bilinijnist asociativnist ta nekomutativnistred Dobutok Kronekera ye chastkovim vipadkom tenzornogo dobutku otzhe vin ye bilinijnim ta asociativnim A B C A B A C displaystyle A otimes B C A otimes B A otimes C nbsp A B C A C B C displaystyle A B otimes C A otimes C B otimes C nbsp k A B A k B k A B displaystyle kA otimes B A otimes kB k A otimes B nbsp A B C A B C displaystyle A otimes B otimes C A otimes B otimes C nbsp de A B ta C ye matricyami a k skalyar dd Dobutok Kronekera ne ye komutativnim Hocha zavzhdi isnuyut taki matrici perestanovki P ta Q sho A B P B A Q displaystyle A otimes B P B otimes A Q nbsp Yaksho A ta B kvadratni matrici todi A displaystyle otimes nbsp B ta B displaystyle otimes nbsp A ye perestanovochno podibnimi tobto P QT A B I B A I displaystyle A otimes B I otimes B A otimes I nbsp de I displaystyle I nbsp odinichna matricya Transponuvannyared Operaciya transponuvannya ye distributivnoyu vidnosno dobutku Kronekera A B T A T B T displaystyle A otimes B T A T otimes B T nbsp Mishanij dobutokred Yaksho A B C ta D ye matricyami takogo rozmiru sho isnuyut dobutki AC ta BD todi A B C D A C B D displaystyle A otimes B C otimes D AC otimes BD nbsp A displaystyle otimes nbsp B ye oborotnoyu todi i tilki todi koli A ta B ye oborotnimi i todi A B 1 A 1 B 1 displaystyle A otimes B 1 A 1 otimes B 1 nbsp Suma ta eksponenta Kronekerared Yaksho A matricya rozmiru n n B matricya rozmiru m m i I k displaystyle I k nbsp odinichna matricya rozmiru k k todi mi mozhemo viznachiti sumu Kronekera displaystyle oplus nbsp yak A B A I m I n B displaystyle A oplus B A otimes I m I n otimes B nbsp Takozh spravedlivo e A B e A e B displaystyle e A oplus B e A otimes e B nbsp Spektr slid ta viznachnikred Yaksho A ta B kvadratni matrici rozmiru n ta q vidpovidno Yaksho l1 ln vlasni znachennya matrici A ta m1 mq vlasni znachennya matrici B Todi vlasnimi znachennyami A displaystyle otimes nbsp B ye l i m j i 1 n j 1 q displaystyle lambda i mu j qquad i 1 ldots n j 1 ldots q nbsp Slid ta viznachnik dobutku Kronekera rivni tr A B tr A tr B displaystyle operatorname tr A otimes B operatorname tr A operatorname tr B nbsp det A B det A q det B n displaystyle det A otimes B det A q det B n nbsp Singulyarnij rozklad ta rangred Yaksho matricya A maye rA nenulovih singulyarnih znachen s A i i 1 r A displaystyle sigma A i qquad i 1 ldots r A nbsp Nenulovi singulyarni znachennya matrici B s B i i 1 r B displaystyle sigma B i qquad i 1 ldots r B nbsp Todi dobutok Kronekera A displaystyle otimes nbsp B maye rArB nenulovih singulyarnih znachen s A i s B j i 1 r A j 1 r B displaystyle sigma A i sigma B j qquad i 1 ldots r A j 1 ldots r B nbsp Rang matrici rivnij kilkosti nenulovih singulyarnih znachen otzhe rank A B rank A rank B displaystyle operatorname rank A otimes B operatorname rank A operatorname rank B nbsp Blokovi versiyi dobutku Kronekerared U vipadku blochnih matric mozhut vikoristovuvatisya operaciyi yaki pov yazani z dobutkom Kronekera odnak vidriznyayutsya poryadkom peremnozhennya blokiv Takimi operaciyami ye dobutok Trejsi Singha angl Tracy Singh product i dobutok Hatri Rao Dobutok Trejsi Singhared Vkazana operaciya mnozhennya blokovih matric polyagaye v tomu sho kozhen blok livoyi matrici mnozhitsya poslidovno na bloki pravoyi matrici Pri comu formuyetsya struktura novoyi matrici yaka vidriznyayetsya vid harakternoyi dlya dobutku Kronekera Dobutok Trejsi Singha viznachayetsya yak 1 2 A B A i j B i j A i j B k l k l i j displaystyle mathbf A circ mathbf B left mathbf A ij circ mathbf B right ij left left mathbf A ij otimes mathbf B kl right kl right ij nbsp Napriklad 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 nbsp A B A 11 B A 12 B A 21 B A 22 B A 11 B 11 A 11 B 12 A 12 B 11 A 12 B 12 A 11 B 21 A 11 B 22 A 12 B 21 A 12 B 22 A 21 B 11 A 21 B 12 A 22 B 11 A 22 B 12 A 21 B 21 A 21 B 22 A 22 B 21 A 22 B 22 1 2 4 7 8 14 3 12 21 4 5 16 28 20 35 6 24 42 2 4 5 8 10 16 6 15 24 3 6 6 9 12 18 9 18 27 8 10 20 32 25 40 12 30 48 12 15 24 36 30 45 18 36 54 7 8 28 49 32 56 9 36 63 14 16 35 56 40 64 18 45 72 21 24 42 63 48 72 27 54 81 displaystyle begin aligned mathbf A circ mathbf B left begin array c c mathbf A 11 circ mathbf B amp mathbf A 12 circ mathbf B hline mathbf A 21 circ mathbf B amp mathbf A 22 circ mathbf B end array right amp left begin array c c c c mathbf A 11 otimes mathbf B 11 amp mathbf A 11 otimes mathbf B 12 amp mathbf A 12 otimes mathbf B 11 amp mathbf A 12 otimes mathbf B 12 hline mathbf A 11 otimes mathbf B 21 amp mathbf A 11 otimes mathbf B 22 amp mathbf A 12 otimes mathbf B 21 amp mathbf A 12 otimes mathbf B 22 hline mathbf A 21 otimes mathbf B 11 amp mathbf A 21 otimes mathbf B 12 amp mathbf A 22 otimes mathbf B 11 amp mathbf A 22 otimes mathbf B 12 hline mathbf A 21 otimes mathbf B 21 amp mathbf A 21 otimes mathbf B 22 amp mathbf A 22 otimes mathbf B 21 amp mathbf A 22 otimes mathbf B 22 end array right amp left begin array c c c c c c c c c 1 amp 2 amp 4 amp 7 amp 8 amp 14 amp 3 amp 12 amp 21 4 amp 5 amp 16 amp 28 amp 20 amp 35 amp 6 amp 24 amp 42 hline 2 amp 4 amp 5 amp 8 amp 10 amp 16 amp 6 amp 15 amp 24 3 amp 6 amp 6 amp 9 amp 12 amp 18 amp 9 amp 18 amp 27 8 amp 10 amp 20 amp 32 amp 25 amp 40 amp 12 amp 30 amp 48 12 amp 15 amp 24 amp 36 amp 30 amp 45 amp 18 amp 36 amp 54 hline 7 amp 8 amp 28 amp 49 amp 32 amp 56 amp 9 amp 36 amp 63 hline 14 amp 16 amp 35 amp 56 amp 40 amp 64 amp 18 amp 45 amp 72 21 amp 24 amp 42 amp 63 amp 48 amp 72 amp 27 amp 54 amp 81 end array right end aligned nbsp Dobutok Hatri Raored Dokladnishe dobutok Hatri Rao Danij variant dobutku viznachenij dlya matric z odnakovoyu blokovoyu strukturoyu Vin peredbachaye sho operaciya kronekerivskogo dobutku vikonuyetsya poblokovo v mezhah odnojmennih matrichnih blokiv za analogiyeyu z poelementnim dobutkom Adamara tilki pri comu v yakosti elementiv zadiyani bloki matric a dlya perenozhennya blokiv vikoristovuyetsya dobutok Kronekera Torcevij dobutokred Dokladnishe torcevij dobutok Vlastivosti mishanih dobutkiv A B C A B C displaystyle mathbf A otimes mathbf B bullet mathbf C mathbf A otimes mathbf B bullet mathbf C nbsp 3 de displaystyle bullet nbsp oznachaye torcevij dobutok 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 4 5 Za analogiyeyu 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 6 de c displaystyle c nbsp i d displaystyle d nbsp vektori 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 7 Analogichno 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 oznachaye vektornu zgortku a F displaystyle mathcal F nbsp ye matriceyu diskretnogo peretvorennya Fur ye 8 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 4 5 de displaystyle ast nbsp oznachaye stovpcevij dobutok Hatri Rao Okrim togo 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 vektori Div takozhred Spisok ob yektiv nazvanih na chest Leopolda Kronekera Dobutok Hatri RaoDzherelared Gantmaher F R Teoriya matric 5 e M Fizmatlit 2010 559 s ISBN 5 9221 0524 8 ros Lankaster P Teoriya matric Moskva Nauka 1973 280 s ros R Horn Ch Dzhonson Matrichnyj analiz M Mir 1989 653 s ros Primitkired Tracy D S Singh R P 1972 A New Matrix Product and Its Applications in Matrix Differentiation Statistica Neerlandica 26 4 143 157 doi 10 1111 j 1467 9574 1972 tb00199 x Liu S 1999 Matrix Results on the Khatri Rao and Tracy Singh Products Linear Algebra and Its Applications 289 1 3 267 277 doi 10 1016 S0024 3795 98 10209 4 Slyusar V I 27 grudnya 1996 End products in matrices in radar applications PDF Radioelectronics and Communications Systems 1998 Vol 41 Number 3 50 53 Arhiv originalu PDF za 27 lipnya 2020 Procitovano 11 serpnya 2020 a b Slyusar V I 13 bereznya 1998 A Family of Face Products of Matrices and its Properties PDF Cybernetics and Systems Analysis C C of Kibernetika I Sistemnyi Analiz 1999 35 3 379 384 doi 10 1007 BF02733426 Arhiv originalu PDF za 25 sichnya 2020 Procitovano 11 serpnya 2020 a b Vadym Slyusar New Matrix Operations for DSP Lecture April 1999 DOI 10 13140 RG 2 2 31620 76164 1 Slyusar V I 15 veresnya 1997 New operations of matrices product for applications of radars PDF Proc Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory DIPED 97 Lviv 73 74 Arhiv originalu PDF za 25 sichnya 2020 Procitovano 11 serpnya 2020 Thomas D Ahle Jakob Baek Tejs Knudsen Almost Optimal Tensor Sketch Published 2019 Mathematics Computer Science ArXiv Arhivovano 28 lipnya 2020 u Wayback Machine Ninh Pham Rasmus Pagh 2013 Fast and scalable polynomial kernels via explicit feature maps SIGKDD international conference on Knowledge discovery and data mining Association for Computing Machinery doi 10 1145 2487575 2487591 Otrimano z https uk wikipedia org w index php title Dobutok Kronekera amp oldid 36887105