Vectorization identity proof
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I'm trying to prove the identity $vert vec(AXB)rangle = Aotimes B^T vert vec(X)rangle$, where $vert vec(L)rangle := sum_{ij} L_{ij}vert iranglevert jrangle$ for any $L:= sum_{ij}L_{ij}vert iranglelangle jvert$.
The left hand side can be written as
$
begin{align}
(AXB)_{in} &= sum_{jpqm}A_{ij}vert iranglelangle jvert X_{pq}vert pranglelangle qvert B_{mn}vert mranglelangle nvert \
&=sum_{jm}A_{ij}X_{jm}B_{mn}vert iranglelangle n vert \
&overset{mathrm{vec}}{=} sum_{jm}A_{ij}X_{jm}B_{mn}vert iranglevert n rangle
end{align}
$,
where I did the vectorization in the last line.
The right hand side is
$
begin{align}
(Aotimes B^T vert vec(X)rangle)_{in} &= sum_{jpqm} A_{ij} vert iranglelangle j vert otimes B_{nm}vert nranglelangle mvert X_{pq} vert prangleotimesvert qrangle \
&= sum_{jm} A_{ij}B_{nm}X_{jm} vert iranglevert n rangle
end{align}
$
These are not equal since $B_{mn} neq B_{nm}$. Can anyone explain what I did wrong?
vectorization
asked Dec 1 at 20:48
user1936752
4951412
add a comment |
up vote
0
down vote
favorite
I'm trying to prove the identity $vert vec(AXB)rangle = Aotimes B^T vert vec(X)rangle$, where $vert vec(L)rangle := sum_{ij} L_{ij}vert iranglevert jrangle$ for any $L:= sum_{ij}L_{ij}vert iranglelangle jvert$.
The left hand side can be written as
$
begin{align}
(AXB)_{in} &= sum_{jpqm}A_{ij}vert iranglelangle jvert X_{pq}vert pranglelangle qvert B_{mn}vert mranglelangle nvert \
&=sum_{jm}A_{ij}X_{jm}B_{mn}vert iranglelangle n vert \
&overset{mathrm{vec}}{=} sum_{jm}A_{ij}X_{jm}B_{mn}vert iranglevert n rangle
end{align}
$,
where I did the vectorization in the last line.
The right hand side is
$
begin{align}
(Aotimes B^T vert vec(X)rangle)_{in} &= sum_{jpqm} A_{ij} vert iranglelangle j vert otimes B_{nm}vert nranglelangle mvert X_{pq} vert prangleotimesvert qrangle \
&= sum_{jm} A_{ij}B_{nm}X_{jm} vert iranglevert n rangle
end{align}
$
These are not equal since $B_{mn} neq B_{nm}$. Can anyone explain what I did wrong?
vectorization
asked Dec 1 at 20:48
user1936752
4951412
You didn't transpose $B$ in the right-hand side.
– Dog_69
Dec 1 at 21:12
Ah right, sorry - silly mistake! I've voted for it to be closed
– user1936752
Dec 1 at 21:17
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm trying to prove the identity $vert vec(AXB)rangle = Aotimes B^T vert vec(X)rangle$, where $vert vec(L)rangle := sum_{ij} L_{ij}vert iranglevert jrangle$ for any $L:= sum_{ij}L_{ij}vert iranglelangle jvert$.
The left hand side can be written as
$
begin{align}
(AXB)_{in} &= sum_{jpqm}A_{ij}vert iranglelangle jvert X_{pq}vert pranglelangle qvert B_{mn}vert mranglelangle nvert \
&=sum_{jm}A_{ij}X_{jm}B_{mn}vert iranglelangle n vert \
&overset{mathrm{vec}}{=} sum_{jm}A_{ij}X_{jm}B_{mn}vert iranglevert n rangle
end{align}
$,
where I did the vectorization in the last line.
The right hand side is
$
begin{align}
(Aotimes B^T vert vec(X)rangle)_{in} &= sum_{jpqm} A_{ij} vert iranglelangle j vert otimes B_{nm}vert nranglelangle mvert X_{pq} vert prangleotimesvert qrangle \
&= sum_{jm} A_{ij}B_{nm}X_{jm} vert iranglevert n rangle
end{align}
$
These are not equal since $B_{mn} neq B_{nm}$. Can anyone explain what I did wrong?
vectorization
asked Dec 1 at 20:48
user1936752
4951412
I'm trying to prove the identity $vert vec(AXB)rangle = Aotimes B^T vert vec(X)rangle$, where $vert vec(L)rangle := sum_{ij} L_{ij}vert iranglevert jrangle$ for any $L:= sum_{ij}L_{ij}vert iranglelangle jvert$.
The left hand side can be written as
$
begin{align}
(AXB)_{in} &= sum_{jpqm}A_{ij}vert iranglelangle jvert X_{pq}vert pranglelangle qvert B_{mn}vert mranglelangle nvert \
&=sum_{jm}A_{ij}X_{jm}B_{mn}vert iranglelangle n vert \
&overset{mathrm{vec}}{=} sum_{jm}A_{ij}X_{jm}B_{mn}vert iranglevert n rangle
end{align}
$,
where I did the vectorization in the last line.
The right hand side is
$
begin{align}
(Aotimes B^T vert vec(X)rangle)_{in} &= sum_{jpqm} A_{ij} vert iranglelangle j vert otimes B_{nm}vert nranglelangle mvert X_{pq} vert prangleotimesvert qrangle \
&= sum_{jm} A_{ij}B_{nm}X_{jm} vert iranglevert n rangle
end{align}
$
These are not equal since $B_{mn} neq B_{nm}$. Can anyone explain what I did wrong?
vectorization
vectorization
asked Dec 1 at 20:48
user1936752
4951412
asked Dec 1 at 20:48
user1936752
4951412
asked Dec 1 at 20:48
user1936752
4951412
asked Dec 1 at 20:48
user1936752
4951412
asked Dec 1 at 20:48
user1936752
4951412
4951412
You didn't transpose $B$ in the right-hand side.
– Dog_69
Dec 1 at 21:12
Ah right, sorry - silly mistake! I've voted for it to be closed
– user1936752
Dec 1 at 21:17
add a comment |
You didn't transpose $B$ in the right-hand side.
– Dog_69
Dec 1 at 21:12
Ah right, sorry - silly mistake! I've voted for it to be closed
– user1936752
Dec 1 at 21:17
You didn't transpose $B$ in the right-hand side.
– Dog_69
Dec 1 at 21:12
You didn't transpose $B$ in the right-hand side.
– Dog_69
Dec 1 at 21:12
Ah right, sorry - silly mistake! I've voted for it to be closed
– user1936752
Dec 1 at 21:17
Ah right, sorry - silly mistake! I've voted for it to be closed
– user1936752
Dec 1 at 21:17
add a comment |
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You didn't transpose $B$ in the right-hand side.
– Dog_69
Dec 1 at 21:12
Ah right, sorry - silly mistake! I've voted for it to be closed
– user1936752
Dec 1 at 21:17