A Wikipédiából, a szabad enciklopédiából
Ez a szócikk vagy szakasz lektorálásra , tartalmi javításokra szorul. A felmerült kifogásokat a szócikk vitalapja részletezi (vagy extrém esetben a szócikk szövegében elhelyezett, kikommentelt szövegrészek ). Ha nincs indoklás a vitalapon (vagy szerkesztési módban a szövegközben), bátran távolítsd el a sablont!
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}}
ij indexű blokk mi pi × nj qj méretű Kronecker-szorzata a megfelelő blokkoknak, feltéve, hogy a két mátrix blokkjainak száma azonos. A szorzat mérete (Σi mi pi j nj qj 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].}
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].}
(
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] 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
[
∗
]
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]}
(
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]
↑ 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 ↑ Zhang X, Yang Z, Cao C. (2002), "Inequalities involving Khatri-Rao products of positive semi-definite matrices", Applied Mathematics E-notes 2 : 117–124 ↑ 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. ↑ 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. ↑ a b c d e f g Slyusar, V. I. (1999. november 23.). „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 . ↑ Slyusar, V. I. (2003. november 23.). „Generalized face-products of matrices in models of digital antenna arrays with nonidentical channels ”. Radioelectronics and Communications Systems 46 (10), 9–17. o. ↑ Anna Esteve, Eva Boj & Josep Fortiana (2009): Interaction Terms in Distance-Based Regression, Communications in Statistics - Theory and Methods, 38:19, P. 3501 [1]
↑ 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. ↑ 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
↑ a b c d e Vadym Slyusar. New Matrix Operations for DSP (Lecture). April 1999. - DOI: 10.13140/RG.2.2.31620.76164/1
![{\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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53cde028551100b310943f25b2028a764436687)
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a086acf8812fc6ff3dec00cf40423b150b9942)
![{\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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/311fb96a2459096ea05d8f0461e67a8b49f5ee43)
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e951f306d0dd52a9a56a35d767f2117db8a5ee6)
![{\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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/760eda68ae4a4ba9e5294dde1c55df20190648c9)
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8842439b57b421cb9c39414d92745f00b00bdae4)
![{\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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86b027efdf1b4ea7866bdc28bc7ac1b3809137bf)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa06f693acbeacb0fa2efcb16170904d4a4735fd)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80fe3459e70e2efe4901d61ec67df754e36798f2)
![{\displaystyle \left(\mathbf {A} [\ast ]\mathbf {B} \right)^{\textsf {T}}={\textbf {A}}^{\textsf {T}}[\bullet ]\mathbf {B} ^{\textsf {T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/907ec4c862586658a29c538a04d6f973e129a19c)