În matematică produsul Khatri–Rao al matricilor este definit drept[ 1] [ 2] [ 3]
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i
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{\displaystyle \mathbf {A} \ast \mathbf {B} =\left(\mathbf {A} _{ij}\otimes \mathbf {B} _{ij}\right)_{ij}}
în care al ij -lea bloc este produsul Kronecker mi pi × nj qj A B (Σi mi pi j nj qj .
De exemplu, dacă ambele A B 2 × 2 :
A
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21
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=
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=
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,
{\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],}
se obține:
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=
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42
14
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72
21
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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].}
Aceasta este o submatrice a produsului Tracy–Singh[ 4]
Produs cu divizarea feței matricilor Un concept alternativ al produsului matricial, care utilizează divizarea pe linii a matricilor cu un anumit număr de linii a fost propus de Vadim Sliusar [ 5] [ 6] [ 7] [ 8] [ 9] [ 10]
Această operație matricială a fost numită „produsul cu divizarea feței” al matricilor[ 7] [ 9]
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=
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,
{\displaystyle \mathbf {C} ={\begin{bmatrix}\mathbf {C} _{1}\\\hline \mathbf {C} _{2}\\\hline \mathbf {C} _{3}\\\end{bmatrix}}={\begin{bmatrix}1&2&3\\\hline 4&5&6\\\hline 7&8&9\end{bmatrix}},\quad \mathbf {D} ={\begin{bmatrix}\mathbf {D} _{1}\\\hline \mathbf {D} _{2}\\\hline \mathbf {D} _{3}\\\end{bmatrix}}={\begin{bmatrix}1&4&7\\\hline 2&5&8\\\hline 3&6&9\end{bmatrix}},}
rezultatul este:[ 6] [ 7] [ 9]
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1
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⊗
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2
C
3
⊗
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=
[
1
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7
2
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14
3
12
21
8
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32
10
25
40
12
30
48
21
42
63
24
48
72
27
54
81
]
.
{\displaystyle \mathbf {C} \bullet \mathbf {D} ={\begin{bmatrix}\mathbf {C} _{1}\otimes \mathbf {D} _{1}\\\hline \mathbf {C} _{2}\otimes \mathbf {D} _{2}\\\hline \mathbf {C} _{3}\otimes \mathbf {D} _{3}\\\end{bmatrix}}={\begin{bmatrix}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{bmatrix}}.}
Transpusa [ 6] [ 7] [ 8]
(
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∙
B
)
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=
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T
∗
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T
{\displaystyle \left(\mathbf {A} \bullet \mathbf {B} \right)^{\textsf {T}}={\textbf {A}}^{\textsf {T}}\ast \mathbf {B} ^{\textsf {T}}}
Biliniaritate și asociativitate :[ 6] [ 7] [ 8]
A
∙
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+
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=
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+
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,
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A
,
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k
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)
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k
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)
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k
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,
(
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=
A
∙
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)
,
{\displaystyle {\begin{aligned}\mathbf {A} \bullet (\mathbf {B} +\mathbf {C} )&=\mathbf {A} \bullet \mathbf {B} +\mathbf {A} \bullet \mathbf {C} ,\\(\mathbf {B} +\mathbf {C} )\bullet \mathbf {A} &=\mathbf {B} \bullet \mathbf {A} +\mathbf {C} \bullet \mathbf {A} ,\\(k\mathbf {A} )\bullet \mathbf {B} &=\mathbf {A} \bullet (k\mathbf {B} )=k(\mathbf {A} \bullet \mathbf {B} ),\\(\mathbf {A} \bullet \mathbf {B} )\bullet \mathbf {C} &=\mathbf {A} \bullet (\mathbf {B} \bullet \mathbf {C} ),\\\end{aligned}}}
unde A , B C k este un scalar ,
a
∙
B
=
B
∙
a
{\displaystyle a\bullet \mathbf {B} =\mathbf {B} \bullet a}
[ 8]
a
{\displaystyle a}
vector ,Proprietatea produsului mixt (V. Sliusar, 1997[ 8]
(
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)
(
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∗
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)
=
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A
A
T
)
∘
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{\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)}
(
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B
)
(
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{\displaystyle (\mathbf {A} \bullet \mathbf {B} )(\mathbf {C} \ast \mathbf {D} )=(\mathbf {A} \mathbf {C} )\circ (\mathbf {B} \mathbf {D} )}
[ 9]
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)
∘
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{\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} )}
[ 11]
(
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T
(
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=
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A
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{\displaystyle (\mathbf {A} \ast \mathbf {B} )^{\textsf {T}}(\mathbf {A} \ast \mathbf {B} )=\left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)\circ \left(\mathbf {B} ^{\textsf {T}}\mathbf {B} \right)}
[ 12]
∘
{\displaystyle \circ }
produsul Hadamard ,
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{\displaystyle (\mathbf {A} \circ \mathbf {B} )\bullet (\mathbf {C} \circ \mathbf {D} )=(\mathbf {A} \bullet \mathbf {C} )\circ (\mathbf {B} \bullet \mathbf {D} )}
[ 8]
a
⊗
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=
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a
⊗
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{\displaystyle \ a\otimes (\mathbf {B} \bullet \mathbf {C} )=(a\otimes \mathbf {B} )\bullet \mathbf {C} }
[ 6]
a
{\displaystyle a}
vector -linie,
(
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⊗
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(
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∗
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=
(
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∗
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{\displaystyle (\mathbf {A} \otimes \mathbf {B} )(\mathbf {C} \ast \mathbf {D} )=(\mathbf {A} \mathbf {C} )\ast (\mathbf {B} \mathbf {D} )}
[ 12]
(
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P
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(
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{\displaystyle (\mathbf {A} \otimes \mathbf {B} )\ast (\mathbf {C} \otimes \mathbf {D} )=\mathbf {P} [(\mathbf {A} \ast \mathbf {C} )\otimes (\mathbf {B} \ast \mathbf {D} )]}
P
{\displaystyle \mathbf {P} }
[ 13]
(
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∙
B
)
(
C
⊗
D
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=
(
A
C
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∙
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{\displaystyle (\mathbf {A} \bullet \mathbf {B} )(\mathbf {C} \otimes \mathbf {D} )=(\mathbf {A} \mathbf {C} )\bullet (\mathbf {B} \mathbf {D} )}
[ 9] [ 11]
(
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L
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⋯
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C
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⋯
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{\displaystyle (\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} )\bullet (\mathbf {L} \mathbf {M} \cdots \mathbf {S} )}
c
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∙
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{\displaystyle c^{\textsf {T}}\bullet d^{\textsf {T}}=c^{\textsf {T}}\otimes d^{\textsf {T}}}
[ 8]
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∗
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=
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⊗
d
{\displaystyle c\ast d=c\otimes d}
c
{\displaystyle c}
d
{\displaystyle d}
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{\displaystyle \left(\mathbf {A} \ast c^{\textsf {T}}\right)d=\left(\mathbf {A} \ast d^{\textsf {T}}\right)c}
[ 14]
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{\displaystyle d^{\textsf {T}}\left(c\bullet \mathbf {A} ^{\textsf {T}}\right)=c^{\textsf {T}}\left(d\bullet \mathbf {A} ^{\textsf {T}}\right)}
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{\displaystyle (\mathbf {A} \bullet \mathbf {B} )(c\otimes d)=(\mathbf {A} c)\circ (\mathbf {B} d)}
[ 15]
c
{\displaystyle c}
d
{\displaystyle d}
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{\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),}
F
(
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{\displaystyle {\mathcal {F}}\left(C^{(1)}x\star C^{(2)}y\right)=\left({\mathcal {F}}C^{(1)}\bullet {\mathcal {F}}C^{(2)}\right)(x\otimes y)={\mathcal {F}}C^{(1)}x\circ {\mathcal {F}}C^{(2)}y}
⋆
{\displaystyle \star }
convoluția vectorilor, iar
F
{\displaystyle {\mathcal {F}}}
matricea Fourier (d
A
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=
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⊗
1
c
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∘
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k
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{\displaystyle \mathbf {A} \bullet \mathbf {B} =\left(\mathbf {A} \otimes \mathbf {1_{c}} ^{\textsf {T}}\right)\circ \left(\mathbf {1_{k}} ^{\textsf {T}}\otimes \mathbf {B} \right)}
[ 16]
A
{\displaystyle \mathbf {A} }
r
×
c
{\displaystyle r\times c}
B
{\displaystyle \mathbf {B} }
r
×
k
{\displaystyle r\times k}
1
c
{\displaystyle \mathbf {1_{c}} }
c
{\displaystyle c}
1
k
{\displaystyle \mathbf {1_{k}} }
k
{\displaystyle k}
M
∙
M
=
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1
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∘
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1
T
⊗
M
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{\displaystyle \mathbf {M} \bullet \mathbf {M} =\left(\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}}\right)\circ \left(\mathbf {1} ^{\textsf {T}}\otimes \mathbf {M} \right)}
[ 17]
M
{\displaystyle \mathbf {M} }
r
×
c
{\displaystyle r\times c}
∘
{\displaystyle \circ }
1
{\displaystyle \mathbf {1} }
c
{\displaystyle c}
M
∙
M
=
M
[
∘
]
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M
⊗
1
T
)
{\displaystyle \mathbf {M} \bullet \mathbf {M} =\mathbf {M} [\circ ]\left(\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}}\right)}
[
∘
]
{\displaystyle [\circ ]}
[ 9]
P
∗
N
=
(
P
⊗
1
c
)
∘
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1
k
⊗
N
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{\displaystyle \mathbf {P} \ast \mathbf {N} =(\mathbf {P} \otimes \mathbf {1_{c}} )\circ (\mathbf {1_{k}} \otimes \mathbf {N} )}
P
{\displaystyle \mathbf {P} }
c
×
r
{\displaystyle c\times r}
N
{\displaystyle \mathbf {N} }
k
×
r
{\displaystyle k\times r}
W
d
A
=
w
∙
A
{\displaystyle \mathbf {W_{d}} \mathbf {A} =\mathbf {w} \bullet \mathbf {A} }
[ 8]
v
e
c
(
(
w
T
∗
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=
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w
{\displaystyle vec((\mathbf {w} ^{\textsf {T}}\ast \mathbf {A} )\mathbf {B} )=(\mathbf {B} ^{\textsf {T}}\ast \mathbf {A} )\mathbf {w} }
[ 9]
v
e
c
(
A
(
w
∙
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)
{\displaystyle vec(\mathbf {A} (\mathbf {w} \bullet \mathbf {B} ))}
vec
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d
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=
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A
∙
A
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T
w
{\displaystyle \operatorname {vec} \left(\mathbf {A} ^{\textsf {T}}\mathbf {W_{d}} \mathbf {A} \right)=\left(\mathbf {A} \bullet \mathbf {A} \right)^{\textsf {T}}\mathbf {w} }
[ 17]
w
{\displaystyle \mathbf {w} }
W
d
{\displaystyle \mathbf {W_{d}} }
vec
(
A
)
{\displaystyle \operatorname {vec} (\mathbf {A} )}
A
{\displaystyle \mathbf {A} }
(
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)
(
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⊗
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)
⋯
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⊗
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)
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=
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.
.
.
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∘
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L
M
.
.
.
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)
{\displaystyle (\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\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} )}
[ 9] [ 11]
(
A
∙
L
)
(
B
⊗
M
)
⋯
(
C
⊗
S
)
(
c
⊗
d
)
=
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A
B
⋯
C
c
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∘
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M
⋯
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d
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,
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)
⋯
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=
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∘
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⋯
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d
)
{\displaystyle {\begin{aligned}(\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )(c\otimes d)&=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} c)\circ (\mathbf {L} \mathbf {M} \cdots \mathbf {S} d),\\(\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )(\mathbf {P} c\otimes \mathbf {Q} d)&=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} \mathbf {P} c)\circ (\mathbf {L} \mathbf {M} \cdots \mathbf {S} \mathbf {Q} d)\end{aligned}}}
c
{\displaystyle c}
d
{\displaystyle d}
(
[
1
0
0
1
1
0
]
∙
[
1
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0
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]
)
(
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−
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]
⊗
[
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−
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]
)
(
[
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⊗
[
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)
(
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∗
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=
(
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)
(
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[
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]
[
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⊗
[
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1
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−
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[
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1
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0
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[
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)
=
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[
1
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]
[
σ
1
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[
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1
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]
∘
[
1
0
1
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0
1
]
[
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1
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−
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]
[
ρ
1
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ρ
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]
[
y
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]
.
{\displaystyle {\begin{aligned}&\left({\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}\bullet {\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}\right)\left({\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\otimes {\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\right)\left({\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}\otimes {\begin{bmatrix}\rho _{1}&0\\0&\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]{}={}&\left({\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}\bullet {\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}\right)\left({\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\,\otimes \,{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\rho _{1}&0\\0&\rho _{2}\\\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}\right)\\[5pt]{}={}&{\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\,\circ \,{\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\rho _{1}&0\\0&\rho _{2}\\\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}.\end{aligned}}}
Produsul cu divizarea feței este utilizat în teoria matricei-tensor a matricei de antene digitale. Aceste operațiuni sunt utilizate și în:
^ en Khatri, C. G.; Rao, C. R. (1968 ). „Solutions to some functional equations and their applications to characterization of probability distributions” . Sankhya . 30 : 167–180. Arhivat din original (PDF) la 23 octombrie 2010 . Accesat în 21 august 2008 . ^ en Liu, Shuangzhe (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 ^ en Zhang X; Yang Z; Cao C. (2002 ), „Inequalities involving Khatri–Rao products of positive semi-definite matrices”, Applied Mathematics E-notes , 2 : 117–124 ^ en
Liu, Shuangzhe; Trenkler, Götz (2008 ). „Hadamard, Khatri-Rao, Kronecker and other matrix products”. International Journal of Information and Systems Sciences . 4 (1): 160–177. ^ en 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 c d e en Slyusar, V. I. (27 decembrie 1996 ). „End products in matrices in radar applications” (PDF) . Izvestiya VUZ: Radioelektronika . 41 (3): 50–53.
^ a b c d e en Slyusar, V. I. (20 mai 1997 ). „Analytical model of the digital antenna array on a basis of face-splitting matrix products” (PDF) . Proc. ICATT-97, Kyiv : 108–109.
^ a b c d e f g h en Slyusar, V. I. (15 septembrie 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.
^ a b c d e f g h en Slyusar, V. I. (13 martie 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 .
^ en Slyusar, V. I. (2003 ). „Generalized face-products of matrices in models of digital antenna arrays with nonidentical channels” (PDF) . Radioelectronics and Communications Systems . 46 (10): 9–17. ^ a b c en Vadym Slyusar. New Matrix Operations for DSP (Lecture). April 1999. – DOI: 10.13140/RG.2.2.31620.76164/1
^ a b en C. R. Rao. Estimation of Heteroscedastic Variances in Linear Models.//Journal of the American Statistical Association, Vol. 65, No. 329 (Mar., 1970), pp. 161–172
^ en Masiero, B.; Nascimento, V. H. (1 mai 2017 ). „Revisiting the Kronecker Array Transform” . IEEE Signal Processing Letters . 24 (5): 525–529. Bibcode :2017ISPL...24..525M . doi :10.1109/LSP.2017.2674969 . ISSN 1070-9908 . ^ en Kasiviswanathan, Shiva Prasad, et al. «The price of privately releasing contingency tables and the spectra of random matrices with correlated rows.» Proceedings of the forty-second ACM symposium on Theory of computing. 2010.^ a b c en Thomas D. Ahle, Jakob Bæk Tejs Knudsen. Almost Optimal Tensor Sketch. Published 2019. Mathematics, Computer Science, ArXiv
^ a b en 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 c en 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 .
^ en Bryan Bischof. Higher order co-occurrence tensors for hypergraphs via face-splitting. Published 15 February 2020, Mathematics, Computer Science, ArXiv ^ en 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]
en Rao C.R.; Rao M. Bhaskara (1998 ), Matrix Algebra and Its Applications to Statistics and Econometrics , World Scientific, p. 216 en Zhang X; Yang Z; Cao C. (2002 ), „Inequalities involving Khatri–Rao products of positive semi-definite matrices”, Applied Mathematics E-notes , 2 : 117–124 en Liu Shuangzhe; Trenkler Götz (2008 ), „Hadamard, Khatri-Rao, Kronecker and other matrix products”, International Journal of Information and Systems Sciences , 4 : 160–177 
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