Khatri–Rao-szorzat

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Mátrixok szorzásánál a Khatri–Rao-szorzat definíciója:[1][2]

A ∗ B = ( A i j ⊗ B i j ) i j {\displaystyle \mathbf {A} \ast \mathbf {B} =(\mathbf {A} _{ij}\otimes \mathbf {B} _{ij})_{ij}} , ahol az ij indexű blokk mipi × njqj méretű Kronecker-szorzata a megfelelő blokkoknak, feltéve, hogy a két mátrix blokkjainak száma azonos. A szorzat mérete (Σi mipi) × (Σj njqj).

Példák:

1. példa:

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}&\mathbf {A} _{12}\\\hline \mathbf {A} _{21}&\mathbf {A} _{22}\end{array}}\right]=\left[{\begin{array}{c c | c}1&2&3\\4&5&6\\\hline 7&8&9\end{array}}\right],\quad \mathbf {B} =\left[{\begin{array}{c | c}\mathbf {B} _{11}&\mathbf {B} _{12}\\\hline \mathbf {B} _{21}&\mathbf {B} _{22}\end{array}}\right]=\left[{\begin{array}{c | c c}1&4&7\\\hline 2&5&8\\3&6&9\end{array}}\right],}
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}&\mathbf {A} _{12}\otimes \mathbf {B} _{12}\\\hline \mathbf {A} _{21}\otimes \mathbf {B} _{21}&\mathbf {A} _{22}\otimes \mathbf {B} _{22}\end{array}}\right]=\left[{\begin{array}{c c | c c}1&2&12&21\\4&5&24&42\\\hline 14&16&45&72\\21&24&54&81\end{array}}\right].}

2. példa: oszloponkénti Khatri–Rao-szorzat:

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}&\mathbf {C} _{2}&\mathbf {C} _{3}\end{array}}\right]=\left[{\begin{array}{c | c | c}1&2&3\\4&5&6\\7&8&9\end{array}}\right],\quad \mathbf {D} =\left[{\begin{array}{c | c | c }\mathbf {D} _{1}&\mathbf {D} _{2}&\mathbf {D} _{3}\end{array}}\right]=\left[{\begin{array}{c | c | c }1&4&7\\2&5&8\\3&6&9\end{array}}\right],}

kapjuk, hogy:

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}&\mathbf {C} _{2}\otimes \mathbf {D} _{2}&\mathbf {C} _{3}\otimes \mathbf {D} _{3}\end{array}}\right]=\left[{\begin{array}{c | c | c }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\end{array}}\right].}

Face-splitting-szorzat

Példák:[3][4][5][6][7]

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&2&3\\\hline 4&5&6\\\hline 7&8&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&4&7\\\hline 2&5&8\\\hline 3&6&9\end{array}}\right],}
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&4&7&2&8&14&3&12&21\\\hline 8&20&32&10&25&40&12&30&48\\\hline 21&42&63&24&48&72&27&54&81\end{array}}\right].}

Tulajdonságai

( 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}}} [4]
( 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)} ,[5][8]
( 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} )} ,[9]

∘ {\displaystyle \circ } - Hadamard-szorzat.

( 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} )} .[8]
( 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} )} [5][9][10]
( 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} )} [5][10]
( 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} )} [5][10]

Block Face-Splitting-szorzat

Transzponált Block Face-Splitting-szorzat[10]

Példák:[3][5]

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}&\mathbf {A} _{12}\\\hline \mathbf {A} _{21}&\mathbf {A} _{22}\end{array}}\right],\quad \mathbf {B} =\left[{\begin{array}{c | c}\mathbf {B} _{11}&\mathbf {B} _{12}\\\hline \mathbf {B} _{21}&\mathbf {B} _{22}\end{array}}\right],}
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}&\mathbf {A} _{12}\bullet \mathbf {B} _{12}\\\hline \mathbf {A} _{21}\bullet \mathbf {B} _{21}&\mathbf {A} _{22}\bullet \mathbf {B} _{22}\end{array}}\right]} .

A transzponált Block Face-Splitting-szorzat[3][5]

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}&\mathbf {A} _{12}\ast \mathbf {B} _{12}\\\hline \mathbf {A} _{21}\ast \mathbf {B} _{21}&\mathbf {A} _{22}\ast \mathbf {B} _{22}\end{array}}\right]} .

Tulajdonságai

( 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}}} [10]

Jegyzetek

  1. ↑ Khatri C. G., C. R. Rao (1968), "Solutions to some functional equations and their applications to characterization of probability distributions", Sankhya 30: 167–180, <http://sankhya.isical.ac.in/search/30a2/30a2019.html>. Hozzáférés ideje: 2020-07-12
  2. ↑ Zhang X, Yang Z, Cao C. (2002), "Inequalities involving Khatri-Rao products of positive semi-definite matrices", Applied Mathematics E-notes 2: 117–124
  3. ↑ a b c Slyusar, V. I. (1996. december 27.). „End products in matrices in radar applications.”. Radioelectronics and Communications Systems.– 1998, Vol. 41; Number 3, 50–53. o.  
  4. ↑ a b Slyusar, V. I. (1997. május 20.). „Analytical model of the digital antenna array on a basis of face-splitting matrix products.”. Proc. ICATT- 97, Kyiv, 108–109. o.  
  5. ↑ a b c d e f g Slyusar, V. I. (1999. július 29.). „A Family of Face Products of Matrices and its Properties”. Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz 35 (3), 379–384. o. DOI:10.1007/BF02733426.  
  6. ↑ Slyusar, V. I. (2003. július 29.). „Generalized face-products of matrices in models of digital antenna arrays with nonidentical channels”. Radioelectronics and Communications Systems 46 (10), 9–17. o.  
  7. ↑ Anna Esteve, Eva Boj & Josep Fortiana (2009): Interaction Terms in Distance-Based Regression, Communications in Statistics - Theory and Methods, 38:19, P. 3501 [1]
  8. ↑ a b Slyusar, V. I. (1997. szeptember 15.). „New operations of matrices product for applications of radars”. Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv., 73–74. o.  
  9. ↑ a b C. Radhakrishna Rao. Estimation of Heteroscedastic Variances in Linear Models.//Journal of the American Statistical Association, Vol. 65, No. 329 (Mar., 1970), pp. 161-172
  10. ↑ a b c d e Vadym Slyusar. New Matrix Operations for DSP (Lecture). April 1999. - DOI: 10.13140/RG.2.2.31620.76164/1

Irodalom

  • Khatri C. G., C. R. Rao (1968). „Solutions to some functional equations and their applications to characterization of probability distributions”. Sankhya 30, 167–180. o. [2010. október 23-i dátummal az eredetiből archiválva]. (Hozzáférés: 2020. július 12.)  
  • 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 & Its Applications to Statistics & Econometrics./C. R. Rao with M. Bhaskara Rao. - World Scientific. - 1998. - P. 216.