Type of product of matrices
In mathematics, the Khatri–Rao product or block Kronecker product of two partitioned matrices A {\displaystyle \mathbf {A} } and B {\displaystyle \mathbf {B} } is defined as[1] [2] [3]
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}} in which the ij -th block is the mi pi × nj qj sized Kronecker product of the corresponding blocks of A and B , assuming the number of row and column partitions of both matrices is equal. The size of the product is then (Σi mi pi ) × (Σj nj qj ) .
For example, if A and B both are 2 × 2 partitioned matrices e.g.:
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],} we obtain:
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].} This is a submatrix of the Tracy–Singh product [4] of the two matrices (each partition in this example is a partition in a corner of the Tracy–Singh product ).
Column-wise Kronecker product [ edit ] The column-wise Kronecker product of two matrices is a special case of the Khatri-Rao product as defined above, and may also be called the Khatri–Rao product. This product assumes the partitions of the matrices are their columns. In this case m 1 = m , p 1 = p , n = q and for each j : nj = pj = 1 . The resulting product is a mp × n matrix of which each column is the Kronecker product of the corresponding columns of A and B . Using the matrices from the previous examples with the columns partitioned:
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],} so that:
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].} This column-wise version of the Khatri–Rao product is useful in linear algebra approaches to data analytical processing[5] and in optimizing the solution of inverse problems dealing with a diagonal matrix.[6] [7]
In 1996 the column-wise Khatri–Rao product was proposed to estimate the angles of arrival (AOAs) and delays of multipath signals[8] and four coordinates of signals sources[9] at a digital antenna array .
Face-splitting product [ edit ] Face splitting product of matrices An alternative concept of the matrix product, which uses row-wise splitting of matrices with a given quantity of rows, was proposed by V. Slyusar [10] in 1996.[9] [11] [12] [13] [14]
This matrix operation was named the "face-splitting product" of matrices[11] [13] or the "transposed Khatri–Rao product". This type of operation is based on row-by-row Kronecker products of two matrices. Using the matrices from the previous examples with the rows partitioned:
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} ={\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}},} the result can be obtained:[9] [11] [13]
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} ={\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}}.} Main properties [ edit ] Transpose (V. Slyusar , 1996[9] [11] [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}}} ,Bilinearity and associativity :[9] [11] [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}}} where A , B and C are matrices, and k is a scalar ,
a ∙ B = B ∙ a {\displaystyle a\bullet \mathbf {B} =\mathbf {B} \bullet a} ,[12] where a {\displaystyle a} is a vector ,The mixed-product property (V. Slyusar , 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} )} ,[13] ( 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} )} [15] ( A ∗ B ) T ( A ∗ B ) = ( A T A ) ∘ ( B T B ) {\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] where ∘ {\displaystyle \circ } denotes the Hadamard product , ( 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 \mathbf {A} \otimes (\mathbf {B} \bullet \mathbf {C} )=(\mathbf {A} \otimes \mathbf {B} )\bullet \mathbf {C} } ,[9] ( 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} )} ,[16] ( A ⊗ B ) ∗ ( C ⊗ D ) = P [ ( A ∗ C ) ⊗ ( B ∗ D ) ] {\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} )]} , where P {\displaystyle \mathbf {P} } is a permutation matrix.[7] ( 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] [15] Similarly: ( A ∙ L ) ( B ⊗ M ) ⋯ ( C ⊗ S ) = ( A B ⋯ C ) ∙ ( L M ⋯ S ) {\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 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} , where c {\displaystyle c} and d {\displaystyle d} are vectors , ( A ∗ c T ) d = ( A ∗ d T ) c {\displaystyle \left(\mathbf {A} \ast c^{\textsf {T}}\right)d=\left(\mathbf {A} \ast d^{\textsf {T}}\right)c} ,[17] d T ( c ∙ A T ) = c T ( d ∙ A T ) {\displaystyle d^{\textsf {T}}\left(c\bullet \mathbf {A} ^{\textsf {T}}\right)=c^{\textsf {T}}\left(d\bullet \mathbf {A} ^{\textsf {T}}\right)} , ( A ∙ B ) ( c ⊗ d ) = ( A c ) ∘ ( B d ) {\displaystyle (\mathbf {A} \bullet \mathbf {B} )(c\otimes d)=(\mathbf {A} c)\circ (\mathbf {B} d)} ,[18] where c {\displaystyle c} and d {\displaystyle d} are vectors (it is a combine of properties 3 an 8), Similarly: ( 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),} F ( C ( 1 ) x ⋆ C ( 2 ) y ) = ( F C ( 1 ) ∙ F C ( 2 ) ) ( x ⊗ y ) = F C ( 1 ) x ∘ F C ( 2 ) y {\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} , where ⋆ {\displaystyle \star } is vector convolution and F {\displaystyle {\mathcal {F}}} is the Fourier transform matrix (this result is an evolving of count sketch properties[19] ), A ∙ B = ( A ⊗ 1 k T ) ∘ ( 1 c T ⊗ B ) {\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] where A {\displaystyle \mathbf {A} } is r × c {\displaystyle r\times c} matrix, B {\displaystyle \mathbf {B} } is r × k {\displaystyle r\times k} matrix, 1 c {\displaystyle \mathbf {1_{c}} } is a vector of 1's of length c {\displaystyle c} , and 1 k {\displaystyle \mathbf {1_{k}} } is a vector of 1's of length k {\displaystyle k} or M ∙ M = ( M ⊗ 1 T ) ∘ ( 1 T ⊗ M ) {\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] where M {\displaystyle \mathbf {M} } is r × c {\displaystyle r\times c} matrix, ∘ {\displaystyle \circ } means element by element multiplication and 1 {\displaystyle \mathbf {1} } is a vector of 1's of length c {\displaystyle c} . M ∙ M = M [ ∘ ] ( M ⊗ 1 T ) {\displaystyle \mathbf {M} \bullet \mathbf {M} =\mathbf {M} [\circ ]\left(\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}}\right)} , where [ ∘ ] {\displaystyle [\circ ]} denotes the penetrating face product of matrices.[13] Similarly: P ∗ N = ( P ⊗ 1 k ) ∘ ( 1 c ⊗ N ) {\displaystyle \mathbf {P} \ast \mathbf {N} =(\mathbf {P} \otimes \mathbf {1_{k}} )\circ (\mathbf {1_{c}} \otimes \mathbf {N} )} , where P {\displaystyle \mathbf {P} } is c × r {\displaystyle c\times r} matrix, N {\displaystyle \mathbf {N} } is k × r {\displaystyle k\times r} matrix,. 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} } [13] = v e c ( A ( w ∙ B ) ) {\displaystyle vec(\mathbf {A} (\mathbf {w} \bullet \mathbf {B} ))} , vec ( A T W d A ) = ( A ∙ A ) 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} } ,[21] where w {\displaystyle \mathbf {w} } is the vector consisting of the diagonal elements of W d {\displaystyle \mathbf {W_{d}} } , vec ( A ) {\displaystyle \operatorname {vec} (\mathbf {A} )} means stack the columns of a matrix A {\displaystyle \mathbf {A} } on top of each other to give a vector. ( 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} )\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] Similarly: ( A ∙ L ) ( B ⊗ M ) ⋯ ( C ⊗ S ) ( c ⊗ d ) = ( A B ⋯ C c ) ∘ ( L M ⋯ S d ) , ( A ∙ L ) ( B ⊗ M ) ⋯ ( C ⊗ S ) ( P c ⊗ Q d ) = ( A B ⋯ C P c ) ∘ ( L M ⋯ S Q 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}}} , where c {\displaystyle c} and d {\displaystyle d} are vectors ( [ 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 0 σ 2 ] [ x 1 x 2 ] ⊗ [ 1 1 1 − 1 ] [ ρ 1 0 0 ρ 2 ] [ y 1 y 2 ] ) = [ 1 0 0 1 1 0 ] [ 1 1 1 − 1 ] [ σ 1 0 0 σ 2 ] [ x 1 x 2 ] ∘ [ 1 0 1 0 0 1 ] [ 1 1 1 − 1 ] [ ρ 1 0 0 ρ 2 ] [ y 1 y 2 ] . {\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}}} If M = T ( 1 ) ∙ ⋯ ∙ T ( c ) {\displaystyle M=T^{(1)}\bullet \dots \bullet T^{(c)}} , where T ( 1 ) , … , T ( c ) {\displaystyle T^{(1)},\dots ,T^{(c)}} are independent components a random matrix T {\displaystyle T} with independent identically distributed rows T 1 , … , T m ∈ R d {\displaystyle T_{1},\dots ,T_{m}\in \mathbb {R} ^{d}} , such that
E [ ( T 1 x ) 2 ] = ‖ x ‖ 2 2 {\displaystyle E\left[(T_{1}x)^{2}\right]=\left\|x\right\|_{2}^{2}} and E [ ( T 1 x ) p ] 1 p ≤ a p ‖ x ‖ 2 {\displaystyle E\left[(T_{1}x)^{p}\right]^{\frac {1}{p}}\leq {\sqrt {ap}}\|x\|_{2}} , then for any vector x {\displaystyle x}
| ‖ M x ‖ 2 − ‖ x ‖ 2 | < ε ‖ x ‖ 2 {\displaystyle \left|\left\|Mx\right\|_{2}-\left\|x\right\|_{2}\right|<\varepsilon \left\|x\right\|_{2}} with probability 1 − δ {\displaystyle 1-\delta } if the quantity of rows
m = ( 4 a ) 2 c ε − 2 log 1 / δ + ( 2 a e ) ε − 1 ( log 1 / δ ) c . {\displaystyle m=(4a)^{2c}\varepsilon ^{-2}\log 1/\delta +(2ae)\varepsilon ^{-1}(\log 1/\delta )^{c}.} In particular, if the entries of T {\displaystyle T} are ± 1 {\displaystyle \pm 1} can get
m = O ( ε − 2 log 1 / δ + ε − 1 ( 1 c log 1 / δ ) c ) {\displaystyle m=O\left(\varepsilon ^{-2}\log 1/\delta +\varepsilon ^{-1}\left({\frac {1}{c}}\log 1/\delta \right)^{c}\right)} which matches the Johnson–Lindenstrauss lemma of m = O ( ε − 2 log 1 / δ ) {\displaystyle m=O\left(\varepsilon ^{-2}\log 1/\delta \right)} when ε {\displaystyle \varepsilon } is small.
Block face-splitting product [ edit ] Transposed block face-splitting product in the context of a multi-face radar model[15] According to the definition of V. Slyusar [9] [13] the block face-splitting product of two partitioned matrices with a given quantity of rows in blocks
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],} can be written as :
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].} The transposed block face-splitting product (or Block column-wise version of the Khatri–Rao product) of two partitioned matrices with a given quantity of columns in blocks has a view:[9] [13]
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].} Main properties [ edit ] Transpose : ( 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}}} [15] Applications [ edit ] The Face-splitting product and the Block Face-splitting product used in the tensor -matrix theory of digital antenna arrays . These operations used also in:
See also [ edit ] ^ Khatri C. G., C. R. Rao (1968). "Solutions to some functional equations and their applications to characterization of probability distributions" . Sankhya . 30 : 167–180. Archived from the original (PDF) on 2010-10-23. Retrieved 2008-08-21 . ^ 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 . ^ Zhang X; Yang Z; Cao C. (2002), "Inequalities involving Khatri–Rao products of positive semi-definite matrices", Applied Mathematics E-notes , 2 : 117–124 ^ Liu, Shuangzhe; Trenkler, Götz (2008). "Hadamard, Khatri-Rao, Kronecker and other matrix products". International Journal of Information and Systems Sciences . 4 (1): 160–177. ^ 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 (2005-01-01). 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