How to rewrite $left[begin{array}{c} Aotimes B_1\ vdots\ Aotimes B_T end{array}right]$?
$begingroup$
We have the following matrix
$left[begin{array}{c}
Aotimes B_1\
vdots\
Aotimes B_T
end{array}right]$, where $A$ and the $B_i$ are matrices, and $otimes$ is the Kronecker product.
Is it possible to rewrite it as a Kronecker product where we gather all the $B_i$ matrices as one big matrix?
For example something like $(cdots) otimes left[begin{array}{c}
B_1\
vdots\
B_T
end{array}right]$
linear-algebra kronecker-product
edited Dec 21 '18 at 13:29
Jean-Claude Arbaut
14.9k63464
asked Dec 21 '18 at 12:42
An old man in the sea.An old man in the sea.
1,65211135
$endgroup$
add a comment |
$begingroup$
We have the following matrix
$left[begin{array}{c}
Aotimes B_1\
vdots\
Aotimes B_T
end{array}right]$, where $A$ and the $B_i$ are matrices, and $otimes$ is the Kronecker product.
Is it possible to rewrite it as a Kronecker product where we gather all the $B_i$ matrices as one big matrix?
For example something like $(cdots) otimes left[begin{array}{c}
B_1\
vdots\
B_T
end{array}right]$
linear-algebra kronecker-product
edited Dec 21 '18 at 13:29
Jean-Claude Arbaut
14.9k63464
asked Dec 21 '18 at 12:42
An old man in the sea.An old man in the sea.
1,65211135
$endgroup$
add a comment |
$begingroup$
We have the following matrix
$left[begin{array}{c}
Aotimes B_1\
vdots\
Aotimes B_T
end{array}right]$, where $A$ and the $B_i$ are matrices, and $otimes$ is the Kronecker product.
Is it possible to rewrite it as a Kronecker product where we gather all the $B_i$ matrices as one big matrix?
For example something like $(cdots) otimes left[begin{array}{c}
B_1\
vdots\
B_T
end{array}right]$
linear-algebra kronecker-product
edited Dec 21 '18 at 13:29
Jean-Claude Arbaut
14.9k63464
asked Dec 21 '18 at 12:42
An old man in the sea.An old man in the sea.
1,65211135
$endgroup$
We have the following matrix
$left[begin{array}{c}
Aotimes B_1\
vdots\
Aotimes B_T
end{array}right]$, where $A$ and the $B_i$ are matrices, and $otimes$ is the Kronecker product.
Is it possible to rewrite it as a Kronecker product where we gather all the $B_i$ matrices as one big matrix?
For example something like $(cdots) otimes left[begin{array}{c}
B_1\
vdots\
B_T
end{array}right]$
linear-algebra kronecker-product
linear-algebra kronecker-product
edited Dec 21 '18 at 13:29
Jean-Claude Arbaut
14.9k63464
asked Dec 21 '18 at 12:42
An old man in the sea.An old man in the sea.
1,65211135
edited Dec 21 '18 at 13:29
Jean-Claude Arbaut
14.9k63464
asked Dec 21 '18 at 12:42
An old man in the sea.An old man in the sea.
1,65211135
edited Dec 21 '18 at 13:29
Jean-Claude Arbaut
14.9k63464
edited Dec 21 '18 at 13:29
Jean-Claude Arbaut
14.9k63464
edited Dec 21 '18 at 13:29
Jean-Claude Arbaut
14.9k63464
14.9k63464
asked Dec 21 '18 at 12:42
An old man in the sea.An old man in the sea.
1,65211135
asked Dec 21 '18 at 12:42
An old man in the sea.An old man in the sea.
1,65211135
asked Dec 21 '18 at 12:42
An old man in the sea.An old man in the sea.
1,65211135
1,65211135
add a comment |
add a comment |
1 Answer
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$begingroup$
You could use the fact that $$mathbf A otimes mathbf B = mathbf K_{M,N} cdot left( mathbf B otimes mathbf Aright) cdot mathbf K_{P,Q}^{rm T},$$ for an $mathbf A in mathbb{F}^{M times P}$, $mathbf B in mathbb{F}^{N times Q}$ and $mathbb{F}$ is the field you are considering.
Here, $mathbf K_{M,N}$ represents the commutation matrix of size $MN times MN$, which is the permutation matrix defined via $$mathbf K_{M,N}cdot {rm vec}{mathbf X} = {rm vec}{mathbf X^{rm T}}$$ for any $mathbf X in mathbb{F}^{M times N}$. A lot is known on these and they have some fun properties. Magnus and Neudecker [1,2] would be a good source to study.
You could apply this to each row to "turn around" all your Kronecker products, which then allows to pull out $mathbf A$ completely. This should give you something like a ${rm diag}{mathbf K_1, ldots, mathbf K_T}$ in front, which may be a bit unhandy. But at least you know that such permutation matrices exist.
Not sure if this helps anything!
[1] Magnus, Jan R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Stat. 7, 381-394 (1979). ZBL0414.62040.
[2] Magnus, Jan R.; Neudecker, Heinz, Matrix differential calculus with applications in statistics and econometrics, Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. Chichester etc.: John Wiley & Sons. XVII, 393 p.; £ 24.50 (1988). ZBL0651.15001.
answered Feb 15 at 12:04
FlorianFlorian
1,5172721
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
oldest
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active
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votes
$begingroup$
You could use the fact that $$mathbf A otimes mathbf B = mathbf K_{M,N} cdot left( mathbf B otimes mathbf Aright) cdot mathbf K_{P,Q}^{rm T},$$ for an $mathbf A in mathbb{F}^{M times P}$, $mathbf B in mathbb{F}^{N times Q}$ and $mathbb{F}$ is the field you are considering.
Here, $mathbf K_{M,N}$ represents the commutation matrix of size $MN times MN$, which is the permutation matrix defined via $$mathbf K_{M,N}cdot {rm vec}{mathbf X} = {rm vec}{mathbf X^{rm T}}$$ for any $mathbf X in mathbb{F}^{M times N}$. A lot is known on these and they have some fun properties. Magnus and Neudecker [1,2] would be a good source to study.
You could apply this to each row to "turn around" all your Kronecker products, which then allows to pull out $mathbf A$ completely. This should give you something like a ${rm diag}{mathbf K_1, ldots, mathbf K_T}$ in front, which may be a bit unhandy. But at least you know that such permutation matrices exist.
Not sure if this helps anything!
[1] Magnus, Jan R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Stat. 7, 381-394 (1979). ZBL0414.62040.
[2] Magnus, Jan R.; Neudecker, Heinz, Matrix differential calculus with applications in statistics and econometrics, Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. Chichester etc.: John Wiley & Sons. XVII, 393 p.; £ 24.50 (1988). ZBL0651.15001.
answered Feb 15 at 12:04
FlorianFlorian
1,5172721
$endgroup$
add a comment |
$begingroup$
You could use the fact that $$mathbf A otimes mathbf B = mathbf K_{M,N} cdot left( mathbf B otimes mathbf Aright) cdot mathbf K_{P,Q}^{rm T},$$ for an $mathbf A in mathbb{F}^{M times P}$, $mathbf B in mathbb{F}^{N times Q}$ and $mathbb{F}$ is the field you are considering.
Here, $mathbf K_{M,N}$ represents the commutation matrix of size $MN times MN$, which is the permutation matrix defined via $$mathbf K_{M,N}cdot {rm vec}{mathbf X} = {rm vec}{mathbf X^{rm T}}$$ for any $mathbf X in mathbb{F}^{M times N}$. A lot is known on these and they have some fun properties. Magnus and Neudecker [1,2] would be a good source to study.
You could apply this to each row to "turn around" all your Kronecker products, which then allows to pull out $mathbf A$ completely. This should give you something like a ${rm diag}{mathbf K_1, ldots, mathbf K_T}$ in front, which may be a bit unhandy. But at least you know that such permutation matrices exist.
Not sure if this helps anything!
[1] Magnus, Jan R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Stat. 7, 381-394 (1979). ZBL0414.62040.
[2] Magnus, Jan R.; Neudecker, Heinz, Matrix differential calculus with applications in statistics and econometrics, Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. Chichester etc.: John Wiley & Sons. XVII, 393 p.; £ 24.50 (1988). ZBL0651.15001.
answered Feb 15 at 12:04
FlorianFlorian
1,5172721
$endgroup$
add a comment |
$begingroup$
You could use the fact that $$mathbf A otimes mathbf B = mathbf K_{M,N} cdot left( mathbf B otimes mathbf Aright) cdot mathbf K_{P,Q}^{rm T},$$ for an $mathbf A in mathbb{F}^{M times P}$, $mathbf B in mathbb{F}^{N times Q}$ and $mathbb{F}$ is the field you are considering.
Here, $mathbf K_{M,N}$ represents the commutation matrix of size $MN times MN$, which is the permutation matrix defined via $$mathbf K_{M,N}cdot {rm vec}{mathbf X} = {rm vec}{mathbf X^{rm T}}$$ for any $mathbf X in mathbb{F}^{M times N}$. A lot is known on these and they have some fun properties. Magnus and Neudecker [1,2] would be a good source to study.
You could apply this to each row to "turn around" all your Kronecker products, which then allows to pull out $mathbf A$ completely. This should give you something like a ${rm diag}{mathbf K_1, ldots, mathbf K_T}$ in front, which may be a bit unhandy. But at least you know that such permutation matrices exist.
Not sure if this helps anything!
[1] Magnus, Jan R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Stat. 7, 381-394 (1979). ZBL0414.62040.
[2] Magnus, Jan R.; Neudecker, Heinz, Matrix differential calculus with applications in statistics and econometrics, Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. Chichester etc.: John Wiley & Sons. XVII, 393 p.; £ 24.50 (1988). ZBL0651.15001.
answered Feb 15 at 12:04
FlorianFlorian
1,5172721
$endgroup$
You could use the fact that $$mathbf A otimes mathbf B = mathbf K_{M,N} cdot left( mathbf B otimes mathbf Aright) cdot mathbf K_{P,Q}^{rm T},$$ for an $mathbf A in mathbb{F}^{M times P}$, $mathbf B in mathbb{F}^{N times Q}$ and $mathbb{F}$ is the field you are considering.
Here, $mathbf K_{M,N}$ represents the commutation matrix of size $MN times MN$, which is the permutation matrix defined via $$mathbf K_{M,N}cdot {rm vec}{mathbf X} = {rm vec}{mathbf X^{rm T}}$$ for any $mathbf X in mathbb{F}^{M times N}$. A lot is known on these and they have some fun properties. Magnus and Neudecker [1,2] would be a good source to study.
You could apply this to each row to "turn around" all your Kronecker products, which then allows to pull out $mathbf A$ completely. This should give you something like a ${rm diag}{mathbf K_1, ldots, mathbf K_T}$ in front, which may be a bit unhandy. But at least you know that such permutation matrices exist.
Not sure if this helps anything!
[1] Magnus, Jan R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Stat. 7, 381-394 (1979). ZBL0414.62040.
[2] Magnus, Jan R.; Neudecker, Heinz, Matrix differential calculus with applications in statistics and econometrics, Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. Chichester etc.: John Wiley & Sons. XVII, 393 p.; £ 24.50 (1988). ZBL0651.15001.
answered Feb 15 at 12:04
FlorianFlorian
1,5172721
answered Feb 15 at 12:04
FlorianFlorian
1,5172721
answered Feb 15 at 12:04
FlorianFlorian
1,5172721
answered Feb 15 at 12:04
FlorianFlorian
1,5172721
1,5172721
add a comment |
add a comment |
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