Произведение Хатри — Рао — операция умножения матриц , определяемая выражением[ 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}}
в котором
i
j
{\displaystyle ij}
произведением Кронекера
m
i
p
i
⊗
n
j
q
j
{\displaystyle m_{i}p_{i}\otimes n_{j}q_{j}}
A
{\displaystyle \mathbf {A} }
B
{\displaystyle \mathbf {B} }
∑
i
m
i
p
i
×
∑
j
n
j
q
j
{\displaystyle \sum _{i}{m_{i}p_{i}}\times \sum _{j}{n_{j}q_{j}}}
К примеру, если матрицы
A
{\displaystyle \mathbf {A} }
B
{\displaystyle \mathbf {B} }
2 × 2 :
A
=
[
A
11
A
12
A
21
A
22
]
=
[
1
2
3
4
5
6
7
8
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]}
B
=
[
B
11
B
12
B
21
B
22
]
=
[
1
4
7
2
5
8
3
6
9
]
{\displaystyle \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]}
то:
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]}
Столбцовое произведение Кронекера двух матриц также принято называть произведением Хатри — Рао.
Это произведение предполагает, что блоки матриц являются их столбцами. В этом случае
m
1
=
m
{\displaystyle m_{1}=m}
p
1
=
p
{\displaystyle p_{1}=p}
n
=
q
{\displaystyle n=q}
j
{\displaystyle j}
n
j
=
p
j
=
1
{\displaystyle n_{j}=p_{j}=1}
m
p
×
n
{\displaystyle mp\times n}
произведение Кронекера соответствующих столбцов матриц
A
{\displaystyle \mathbf {A} }
B
{\displaystyle \mathbf {B} }
C
=
[
C
1
C
2
C
3
]
=
[
1
2
3
4
5
6
7
8
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]}
D
=
[
D
1
D
2
D
3
]
=
[
1
4
7
2
5
8
3
6
9
]
{\displaystyle \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]}
столбцовое произведение:
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]}
Столбцовая версия произведения Хатри — Рао используется в линейной алгебре для аналитической обработки данных[ 3] [ 4] [ 5] цифровой антенной решётке [ 6] радара [ 7]
Торцевое произведение матриц Существует альтернативная концепция произведения матриц, которая в отличие от столбцовой версии использует разбиение матриц на строки[ 8] торцевое произведение (англ. face-splitting product )[ 7] [ 9] [ 10] транспонированное произведение Хатри — Рао (англ. transposed Khatri — Rao product )[ 11]
C
=
[
C
1
C
2
C
3
]
=
[
1
2
3
4
5
6
7
8
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]}
D
=
[
D
1
D
2
D
3
]
=
[
1
4
7
2
5
8
3
6
9
]
{\displaystyle \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]}
можно записать[ 7]
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]}
Транспонирование (1996[ 7] [ 9] [ 12]
(
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}}}
Коммутативность и ассоциативная операция [ 7] [ 9] [ 12]
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} )&=\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}}}
где
A
{\displaystyle \mathbf {A} }
A
{\displaystyle \mathbf {A} }
C
{\displaystyle \mathbf {C} }
k
{\displaystyle k}
a
∙
B
=
B
∙
a
{\displaystyle a\bullet \mathbf {B} =\mathbf {B} \bullet a}
[ 12]
a
{\displaystyle a}
B
{\displaystyle \mathbf {B} }
Свойство смешанного произведения (1997[ 12]
(
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)}
(
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} )}
[ 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} )}
[ 11] [ 13]
(
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} )}
[ 14] где
∘
{\displaystyle \circ }
произведение Адамара .
Также выполняются следующие свойства:
(
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} )}
[ 12]
a
⊗
(
B
∙
C
)
=
(
a
⊗
B
)
∙
C
{\displaystyle \ a\otimes (\mathbf {B} \bullet \mathbf {C} )=(a\otimes \mathbf {B} )\bullet \mathbf {C} }
[ 7]
a
{\displaystyle a}
(
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} )}
[ 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} )}
[ 13]
(
A
∙
L
)
(
B
⊗
M
)
…
(
C
⊗
S
)
=
(
A
B
…
C
)
∙
(
L
M
.
.
.
S
)
{\displaystyle (\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\dots (\mathbf {C} \otimes \mathbf {S} )=(\mathbf {A} \mathbf {B} \dots \mathbf {C} )\bullet (\mathbf {L} \mathbf {M} ...\mathbf {S} )}
c
T
∙
d
T
=
c
T
⊗
d
T
{\displaystyle c^{\textsf {T}}\bullet d^{\textsf {T}}=c^{\textsf {T}}\otimes d^{\textsf {T}}}
[ 12]
c
∗
d
=
c
⊗
d
{\displaystyle c\ast d=c\otimes d}
c
{\displaystyle c}
d
{\displaystyle d}
(
A
∗
c
T
)
d
=
(
A
∗
d
T
)
c
{\displaystyle (\mathbf {A} \ast c^{\textsf {T}})d=(\mathbf {A} \ast d^{\textsf {T}})c}
[ 15]
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}})}
(
A
∙
B
)
(
c
⊗
d
)
=
(
A
c
)
∘
(
B
d
)
{\displaystyle (\mathbf {A} \bullet \mathbf {B} )(c\otimes d)=(\mathbf {A} c)\circ (\mathbf {B} d)}
[ 16]
c
{\displaystyle c}
d
{\displaystyle d}
(
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)}
W
(
C
(
1
)
x
⋆
C
(
2
)
y
)
=
(
W
C
(
1
)
∙
W
C
(
2
)
)
(
x
⊗
y
)
=
W
C
(
1
)
x
∘
W
C
(
2
)
y
{\displaystyle {\mathcal {W}}(C^{(1)}x\star C^{(2)}y)=({\mathcal {W}}C^{(1)}\bullet {\mathcal {W}}C^{(2)})(x\otimes y)={\mathcal {W}}C^{(1)}x\circ {\mathcal {W}}C^{(2)}y}
где
W
{\displaystyle {\mathcal {W}}}
преобразования Фурье ,
⋆
{\displaystyle \star }
свёртки (тождество следует из свойств отсчётного скетча [ 17]
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} )}
[ 18]
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}} }
1
k
{\displaystyle \mathbf {1_{k}} }
c
{\displaystyle c}
k
{\displaystyle k}
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} )}
[ 19]
M
{\displaystyle \mathbf {M} }
r
×
c
{\displaystyle r\times c}
∘
{\displaystyle \circ }
произведение Адамара и
1
{\displaystyle \mathbf {1} }
c
{\displaystyle c}
M
∙
M
=
M
[
∘
]
(
M
⊗
1
T
)
{\displaystyle \mathbf {M} \bullet \mathbf {M} =\mathbf {M} [\circ ](\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}})}
[
∘
]
{\displaystyle [\circ ]}
проникающего торцевого произведения матриц.По аналогии,
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} )}
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} }
[ 12]
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} }
[ 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} }
[ 11]
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} }
[ 19]
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} }
где
w
{\displaystyle \mathbf {w} }
W
d
{\displaystyle \mathbf {W_{d}} }
v
e
c
(
A
)
{\displaystyle vec(\mathbf {A} )}
A
{\displaystyle \mathbf {A} }
Свойство поглощения произведения Кронекера:
(
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} )}
[ 10] [ 13]
(
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)}
(
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)}
где
c
{\displaystyle c}
d
{\displaystyle d}
Например[ 16]
(
[
1
0
0
1
1
0
]
∙
[
1
0
1
0
0
1
]
)
(
[
1
1
1
−
1
]
⊗
[
1
1
1
−
1
]
)
(
[
σ
1
0
0
σ
2
]
⊗
[
ρ
1
0
0
ρ
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
]
[
σ
1
0
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{\displaystyle {\begin{aligned}\\&\quad \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]&\quad =\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]&\quad ={\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}}}
Если
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леммы Джонсона-Линденштрауса о распределении.
Примение блочного транспонированного торцевого произведения для описания отклика многогранной цифровой антенной решётки [ 13] Для блочных матриц с одинаковым количеством столбцов в соответствующих блоках:
A
=
[
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A
12
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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]}
B
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B
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B
12
B
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B
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{\displaystyle \mathbf {B} =\left[{\begin{array}{c | c}\mathbf {B} _{11}&\mathbf {B} _{12}\\\hline \mathbf {B} _{21}&\mathbf {B} _{22}\end{array}}\right]}
согласно определению[ 7]
A
[
∙
]
B
{\displaystyle \mathbf {A} [\bullet ]\mathbf {B} }
A
[
∙
]
B
=
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11
∙
B
11
A
12
∙
B
12
A
21
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A
22
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{\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]}
Аналогично, для блочного транспонированного торцевого произведения (или блочного столбцового произведения Хатри — Рао ) двух матриц
A
[
∗
]
B
{\displaystyle \mathbf {A} [\ast ]\mathbf {B} }
[ 7]
A
[
∗
]
B
=
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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]}
Выполняется свойство транспонирования[ 13]
(
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}}}
Семейство торцевых произведений матриц используется в тензорно-матричной теории цифровых антенных решёток для радиотехнических систем[ 11]
Торцевое произведение получило широкое распространение в системах машинного обучения, статистической обработке больших данных[ 16] тензорный скетч [ 16] быстрого преобразования Джонсона — Линденштрауса [ 16] произведению Адамара , оперирующему матрицами меньшей размерности. Погрешность аппроксимации данных большой размерности на основе торцевого произведения матриц соответствует лемме о малом искажении [ 16] [ 20] ➤ дифференциальной приватности (англ. differential privacy )[ 15] обработки естественного языка и построения гиперграфов подобия изображений[ 21]
Торцевое произведение применяется для P-сплайн аппроксимации[ 18] [ 19] ядерного метода машинного обучения , а также изучения взаимодействия генотипов с окружающей средой.[ 22]
↑ Khatri C. G., C. R. Rao . Solutions to some functional equations and their applications to characterization of probability distributions (англ.) // Sankhya [англ.] Vol. 30 . — P. 167—180 . Архивировано 23 октября 2010 года.↑ 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. Efficient Solution of Linear Matrix Equations with Application to Multistatic Antenna Array Processing (EN) // Communications in Information & Systems. — 2005. — 1 января (т. 05 , № 1 ). — С. 123—130 . — ISSN 1526-7555 . — doi :10.4310/CIS.2005.v5.n1.a5 . Архивировано 12 июля 2020 года.↑ Masiero, B.; Nascimento, V. H. Revisiting the Kronecker Array Transform // IEEE Signal Processing Letters. — 2017. — 1 мая (т. 24 , № 5 ). — С. 525—529 . — ISSN 1070-9908 . — doi :10.1109/LSP.2017.2674969 . — Bibcode : 2017ISPL...24..525M Архивировано 12 июля 2020 года.↑ Vanderveen, M. C., Ng, B. C., Papadias, C. B., & 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
↑ 1 2 3 4 5 6 7 8 Slyusar, V. I. (December 27, 1996). "End products in matrices in radar applications" (PDF) . Izvestiya VUZ: Radioelektronika.– 1998, Vol. 41; Number 3 : 50—53. Архивировано (PDF) 27 июля 2020 . Дата обращения: 27 июля 2020 . ↑ Anna Esteve, Eva Boj & Josep Fortiana (2009): Interaction Terms in Distance-Based Regression, Communications in Statistics — Theory and Methods, 38:19, P. 3501 [1] Архивная копия от 26 апреля 2021 на Wayback Machine
↑ 1 2 3 Slyusar, V. I. Analytical model of the digital antenna array on a basis of face-splitting matrix products (англ.) // Proc. ICATT- 97, Kyiv : journal. — 1997. — 20 May. — P. 108—109 . Архивировано 25 января 2020 года.↑ 1 2 3 4 Slyusar, V. I. (1999). "A Family of Face Products of Matrices and its Properties" (PDF) . Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz . 35 (3): 379—384. doi :10.1007/BF02733426 . Архивировано (PDF) 25 января 2020 . Дата обращения: 12 июля 2020 . ↑ 1 2 3 4 Миночкин А. И., Рудаков В. И., Слюсар В. И. Основы военно-технических исследований. Теория и приложения. Том. 2. Синтез средств информационного обеспечения вооружения и военной техники // Под ред. А. П. Ковтуненко. — Киев: «Гранмна». — 2012. (неопр.) C. 7 - 98; 354 - 521 (2012). Дата обращения: 12 июля 2020. Архивировано 25 января 2020 года.↑ 1 2 3 4 5 6 7 Slyusar, V. I. (1997-09-15). "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. Архивировано (PDF) 25 января 2020 . Дата обращения: 12 июля 2020 . ↑ 1 2 3 4 5 Vadym Slyusar. New Matrix Operations for DSP (Lecture). April 1999. - DOI: 10.13140/RG.2.2.31620.76164/1
↑ 1 2 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↑ 1 2 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.
↑ 1 2 3 4 5 6 7 Ahle, Thomas; Knudsen, Jakob Almost Optimal Tensor Sketch (неопр.) . [2] Архивировано 14 июля 2020 года.↑ 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 . ↑ 1 2 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 . ↑ 1 2 3 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 Архивная копия от 25 ноября 2020 на 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. [3]
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00683212cf2ac381d32afbd91292c2eed3935d3b)
![{\displaystyle \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/619203aae2a43f2533df3ddb1cf90e08e8b0fb35)
![{\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/f9fc15d35c58601d035fa791a6dfd5b88067ab71)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b179f177e0bc2e894bd939311f9b1e39cd744284)
![{\displaystyle \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/48c6be4fd9b2e7573173b58fba51877ddde58980)
![{\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/eee3b103ab04318c0e74efb8283944f4f8996698)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c17fe205d2885dab091852f046b6c14e8df77e5)
![{\displaystyle \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/16be1c4ae0ca987d31a3648dd444526bdcf4a4df)
![{\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/908c3246daae192cd8e2a796daf7083770bdc170)
}](https://wikimedia.org/api/rest_v1/media/math/render/svg/861e013501c6602cb57058543bfe9ca9f409492d)
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![{\displaystyle {\begin{aligned}\\&\quad \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]&\quad =\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]&\quad ={\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b41b6013987f68a19d76fdfa669fca9462076c24)
![{\displaystyle E[(T_{1}x)^{2}]=\|x\|_{2}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2314df29d3e99c399b8fca4812bf182de141ef4a)
![{\displaystyle E[(T_{1}x)^{p}]^{1/p}\leq {\sqrt {ap}}\|x\|_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b322fc4b1f87234e363acf27136664c4097d2af)
![{\displaystyle \mathbf {A} =\left[{\begin{array}{c | c}\mathbf {A} _{11}&\mathbf {A} _{12}\\\hline \mathbf {A} _{21}&\mathbf {A} _{22}\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae9951e1a02733640a5df02a42796d5b0b73760a)
![{\displaystyle \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/8782967d3ca3d31a089f35809b7586a1f29ddd9c)
![{\displaystyle \mathbf {A} [\bullet ]\mathbf {B} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/67a135d00e547772dad41d1ff38e39f10153d815)
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8789652968356a415c58acf55f2b1d6695a36f80)
![{\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)