Producto Khatri-Rao

${\displaystyle \left(\mathbf {A} \bullet \mathbf {B} \right)^{\textsf {T}}={\textbf {A}}^{\textsf {T}}\ast \mathbf {B} ^{\textsf {T}}}$,

${\displaystyle {\begin{aligned}\mathbf {A} \bullet (\mathbf {B} +\mathbf {C} )&=\mathbf {A} \bullet \mathbf {B} +\mathbf {A} \bullet \mathbf {C} ,\\(\mathbf {B} +\mathbf {C} )\bullet \mathbf {A} &=\mathbf {B} \bullet \mathbf {A} +\mathbf {C} \bullet \mathbf {A} ,\\(k\mathbf {A} )\bullet \mathbf {B} &=\mathbf {A} \bullet (k\mathbf {B} )=k(\mathbf {A} \bullet \mathbf {B} ),\\(\mathbf {A} \bullet \mathbf {B} )\bullet \mathbf {C} &=\mathbf {A} \bullet (\mathbf {B} \bullet \mathbf {C} ),\\\end{aligned}}}$

donde A , B y C son matrices, y k es un escalar ,

${\displaystyle a\bullet \mathbf {B} =\mathbf {B} \bullet a}$, ^{[ 12 ]}

${\displaystyle a}$

${\displaystyle (\mathbf {A} \bullet \mathbf {B} )\left(\mathbf {A} ^{\textsf {T}}\ast \mathbf {B} ^{\textsf {T}}\right)=\left(\mathbf {A} \mathbf {A} ^{\textsf {T}}\right)\circ \left(\mathbf {B} \mathbf {B} ^{\textsf {T}}\right)}$,

${\displaystyle (\mathbf {A} \bullet \mathbf {B} )(\mathbf {C} \ast \mathbf {D} )=(\mathbf {A} \mathbf {C} )\circ (\mathbf {B} \mathbf {D} )}$, ^{[ 13 ]}

${\displaystyle (\mathbf {A} \bullet \mathbf {B} \bullet \mathbf {C} \bullet \mathbf {D} )(\mathbf {L} \ast \mathbf {M} \ast \mathbf {N} \ast \mathbf {P} )=(\mathbf {A} \mathbf {L} )\circ (\mathbf {B} \mathbf {M} )\circ (\mathbf {C} \mathbf {N} )\circ (\mathbf {D} \mathbf {P} )}$^{[ 15 ]}

${\displaystyle (\mathbf {A} \ast \mathbf {B} )^{\textsf {T}}(\mathbf {A} \ast \mathbf {B} )=\left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)\circ \left(\mathbf {B} ^{\textsf {T}}\mathbf {B} \right)}$, ^{[ 16 ]}

${\displaystyle \circ }$

${\displaystyle (\mathbf {A} \bullet \mathbf {B} )(\mathbf {C} \otimes \mathbf {D} )=(\mathbf {A} \mathbf {C} )\bullet (\mathbf {B} \mathbf {D} )}$, ^{[ 13 ] [ 15 ]}

Similarmente:

${\displaystyle (\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} )\bullet (\mathbf {L} \mathbf {M} \cdots \mathbf {S} )}$,

${\displaystyle c^{\textsf {T}}\bullet d^{\textsf {T}}=c^{\textsf {T}}\otimes d^{\textsf {T}}}$, ^{[ 12 ]}

${\displaystyle c\ast d=c\otimes d}$,

${\displaystyle c}$

${\displaystyle d}$

${\displaystyle (\mathbf {A} \bullet \mathbf {B} )(c\otimes d)=(\mathbf {A} c)\circ (\mathbf {B} d)}$, ^{[ 18 ]}

dónde${\displaystyle c}$y${\displaystyle d}$son vectores (es una combinación de las propiedades 3 y 8),

Similarmente:

${\displaystyle (\mathbf {A} \bullet \mathbf {B} )(\mathbf {M} \mathbf {N} c\otimes \mathbf {Q} \mathbf {P} d)=(\mathbf {A} \mathbf {M} \mathbf {N} c)\circ (\mathbf {B} \mathbf {Q} \mathbf {P} d),}$

${\displaystyle {\mathcal {F}}\left((C^{(1)}x)\star (C^{(2)}y)\right)=\left(({\mathcal {F}}C^{(1)})\bullet ({\mathcal {F}}C^{(2)})\right)(x\otimes y)=({\mathcal {F}}C^{(1)}x)\circ ({\mathcal {F}}C^{(2)}y)}$,

dónde${\displaystyle \star }$es convolución vectorial ;${\displaystyle C^{(1)},C^{(2)}}$son matrices de "recuento de bocetos"; y${\displaystyle {\mathcal {F}}}$es la matriz de transformada de Fourier (este resultado es una evolución de las propiedades del boceto de conteo ^{[ 19 ]} ).

Esto puede generalizarse para matrices apropiadas.${\displaystyle \mathbf {A} ,\mathbf {B} }$:

${\displaystyle {\mathcal {F}}\left((\mathbf {A} x)\star (\mathbf {B} y)\right)=\left(({\mathcal {F}}\mathbf {A} )\bullet ({\mathcal {F}}\mathbf {B} )\right)(x\otimes y)=({\mathcal {F}}\mathbf {A} x)\circ ({\mathcal {F}}\mathbf {B} y)}$

porque la propiedad 11 anterior nos da

${\displaystyle \left(({\mathcal {F}}\mathbf {A} )\bullet ({\mathcal {F}}\mathbf {B} )\right)(x\otimes y)=({\mathcal {F}}\mathbf {A} x)\circ ({\mathcal {F}}\mathbf {B} y)}$

Y el teorema de convolución nos da

${\displaystyle {\mathcal {F}}\left((\mathbf {A} x)\star (\mathbf {B} y)\right)=({\mathcal {F}}\mathbf {A} x)\circ ({\mathcal {F}}\mathbf {B} y)}$

${\displaystyle \mathbf {A} \bullet \mathbf {B} =\left(\mathbf {A} \otimes \mathbf {1_{k}} ^{\textsf {T}}\right)\circ \left(\mathbf {1_{c}} ^{\textsf {T}}\otimes \mathbf {B} \right)}$, ^{[ 20 ]}

dónde${\displaystyle \mathbf {A} }$es${\displaystyle r\times c}$matriz,${\displaystyle \mathbf {B} }$es${\displaystyle r\times k}$matriz,${\displaystyle \mathbf {1_{c}} }$es un vector de 1 de longitud${\displaystyle c}$, y${\displaystyle \mathbf {1_{k}} }$es un vector de 1 de longitud${\displaystyle k}$

o

${\displaystyle \mathbf {M} \bullet \mathbf {M} =\left(\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}}\right)\circ \left(\mathbf {1} ^{\textsf {T}}\otimes \mathbf {M} \right)}$, ^{[ 21 ]}

dónde${\displaystyle \mathbf {M} }$es${\displaystyle r\times c}$matriz,${\displaystyle \circ }$significa multiplicación elemento por elemento y${\displaystyle \mathbf {1} }$es un vector de 1 de longitud${\displaystyle c}$.

${\displaystyle \mathbf {M} \bullet \mathbf {M} =\mathbf {M} [\circ ]\left(\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}}\right)}$,

dónde${\displaystyle [\circ ]}$denota el producto de caras penetrantes de matrices. ^{[ 13 ]}

Similarmente:

${\displaystyle \mathbf {P} \ast \mathbf {N} =(\mathbf {P} \otimes \mathbf {1_{k}} )\circ (\mathbf {1_{c}} \otimes \mathbf {N} )}$, dónde${\displaystyle \mathbf {P} }$es${\displaystyle c\times r}$matriz,${\displaystyle \mathbf {N} }$es${\displaystyle k\times r}$matriz,.

${\displaystyle \mathbf {W_{d}} \mathbf {A} =\mathbf {w} \bullet \mathbf {A} }$, ^{[ 12 ]}

${\displaystyle \operatorname {vec} ((\mathbf {w} ^{\textsf {T}}\ast \mathbf {A} )\mathbf {B} )=(\mathbf {B} ^{\textsf {T}}\ast \mathbf {A} )\mathbf {w} }$^{[ 13 ]} =${\displaystyle \operatorname {vec} (\mathbf {A} (\mathbf {w} \bullet \mathbf {B} ))}$=${\displaystyle \operatorname {vec} (\mathbf {A} \mathbf {W_{d}} \mathbf {B} )}$,

${\displaystyle \operatorname {vec} \left(\mathbf {A} ^{\textsf {T}}\mathbf {W_{d}} \mathbf {A} \right)=\left(\mathbf {A} \bullet \mathbf {A} \right)^{\textsf {T}}\mathbf {w} }$, ^{[ 21 ]}

${\displaystyle \mathbf {w} }$

${\displaystyle \mathbf {W_{d}} }$

${\displaystyle \operatorname {vec} (\mathbf {A} )}$

${\displaystyle \mathbf {A} }$

${\displaystyle (\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )(\mathbf {K} \ast \mathbf {T} )=(\mathbf {A} \mathbf {B} ...\mathbf {C} \mathbf {K} )\circ (\mathbf {L} \mathbf {M} ...\mathbf {S} \mathbf {T} )}$. ^{[ 13 ] [ 15 ]}

Similarmente:

${\displaystyle {\begin{aligned}(\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )(c\otimes d)&=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} c)\circ (\mathbf {L} \mathbf {M} \cdots \mathbf {S} d),\\(\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )(\mathbf {P} c\otimes \mathbf {Q} d)&=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} \mathbf {P} c)\circ (\mathbf {L} \mathbf {M} \cdots \mathbf {S} \mathbf {Q} d)\end{aligned}}}$,

${\displaystyle c}$

${\displaystyle d}$

Si${\textstyle \mathbf {B} }$es una matriz diagonal y${\textstyle \mathbf {b} }$es su diagonal principal:

${\displaystyle \operatorname {vec} (\mathbf {A} \mathbf {B} \mathbf {C} )=(\mathbf {C} ^{\mathrm {T} }\ast \mathbf {A} )\mathbf {b} }$

${\textstyle \operatorname {vec} (\cdot )}$