tensor product of two column stochastic matrix [closed]
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Is the tensor product of two column/row stochastic matrix is again a column/row stochastic?
Thanks for helping.
matrices stochastic-matrices
asked Nov 30 '18 at 11:44
MarkovMarkov
17.3k1059178
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closed as off-topic by amWhy, user10354138, Markov, Brahadeesh, Henrik Nov 30 '18 at 14:32
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- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – user10354138, Brahadeesh, Henrik
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$begingroup$
Is the tensor product of two column/row stochastic matrix is again a column/row stochastic?
Thanks for helping.
matrices stochastic-matrices
asked Nov 30 '18 at 11:44
MarkovMarkov
17.3k1059178
$endgroup$
closed as off-topic by amWhy, user10354138, Markov, Brahadeesh, Henrik Nov 30 '18 at 14:32
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – user10354138, Brahadeesh, Henrik
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Is the tensor product of two column/row stochastic matrix is again a column/row stochastic?
Thanks for helping.
matrices stochastic-matrices
asked Nov 30 '18 at 11:44
MarkovMarkov
17.3k1059178
$endgroup$
Is the tensor product of two column/row stochastic matrix is again a column/row stochastic?
Thanks for helping.
matrices stochastic-matrices
matrices stochastic-matrices
asked Nov 30 '18 at 11:44
MarkovMarkov
17.3k1059178
asked Nov 30 '18 at 11:44
MarkovMarkov
17.3k1059178
edited Dec 1 '18 at 17:31
Markov
asked Nov 30 '18 at 11:44
MarkovMarkov
17.3k1059178
asked Nov 30 '18 at 11:44
MarkovMarkov
17.3k1059178
asked Nov 30 '18 at 11:44
MarkovMarkov
17.3k1059178
17.3k1059178
closed as off-topic by amWhy, user10354138, Markov, Brahadeesh, Henrik Nov 30 '18 at 14:32
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – user10354138, Brahadeesh, Henrik
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by amWhy, user10354138, Markov, Brahadeesh, Henrik Nov 30 '18 at 14:32
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – user10354138, Brahadeesh, Henrik
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
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1 Answer
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$begingroup$
The tensor product of two $ntimes n$ matrices $A,B$ is
$$Aotimes B = left(begin{array}{ccc} a_{11}B & ldots& a_{1n}B\
vdots & ldots & vdots\
a_{n1}B & ldots & a_{nn} B
end{array}right).$$
So if $A,B$ are column stochastic, then so is $Aotimes B$. Just check the $j$-th column of the block matrix $Aotimes B$ and take inside the block matrix the $k$-th column of $B$:
$$a_{1j}[b_{1k}+ldots+b_{nk}] + ldots + a_{nj}[b_{1k}+ldots+b_{nk}] = a_{1j}+ldots+a_{nj} = 1.$$
Similar for the rows.
answered Nov 30 '18 at 11:52
WuestenfuxWuestenfux
4,4981413
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The tensor product of two $ntimes n$ matrices $A,B$ is
$$Aotimes B = left(begin{array}{ccc} a_{11}B & ldots& a_{1n}B\
vdots & ldots & vdots\
a_{n1}B & ldots & a_{nn} B
end{array}right).$$
So if $A,B$ are column stochastic, then so is $Aotimes B$. Just check the $j$-th column of the block matrix $Aotimes B$ and take inside the block matrix the $k$-th column of $B$:
$$a_{1j}[b_{1k}+ldots+b_{nk}] + ldots + a_{nj}[b_{1k}+ldots+b_{nk}] = a_{1j}+ldots+a_{nj} = 1.$$
Similar for the rows.
answered Nov 30 '18 at 11:52
WuestenfuxWuestenfux
4,4981413
$endgroup$
add a comment |
$begingroup$
The tensor product of two $ntimes n$ matrices $A,B$ is
$$Aotimes B = left(begin{array}{ccc} a_{11}B & ldots& a_{1n}B\
vdots & ldots & vdots\
a_{n1}B & ldots & a_{nn} B
end{array}right).$$
So if $A,B$ are column stochastic, then so is $Aotimes B$. Just check the $j$-th column of the block matrix $Aotimes B$ and take inside the block matrix the $k$-th column of $B$:
$$a_{1j}[b_{1k}+ldots+b_{nk}] + ldots + a_{nj}[b_{1k}+ldots+b_{nk}] = a_{1j}+ldots+a_{nj} = 1.$$
Similar for the rows.
answered Nov 30 '18 at 11:52
WuestenfuxWuestenfux
4,4981413
$endgroup$
add a comment |
$begingroup$
The tensor product of two $ntimes n$ matrices $A,B$ is
$$Aotimes B = left(begin{array}{ccc} a_{11}B & ldots& a_{1n}B\
vdots & ldots & vdots\
a_{n1}B & ldots & a_{nn} B
end{array}right).$$
So if $A,B$ are column stochastic, then so is $Aotimes B$. Just check the $j$-th column of the block matrix $Aotimes B$ and take inside the block matrix the $k$-th column of $B$:
$$a_{1j}[b_{1k}+ldots+b_{nk}] + ldots + a_{nj}[b_{1k}+ldots+b_{nk}] = a_{1j}+ldots+a_{nj} = 1.$$
Similar for the rows.
answered Nov 30 '18 at 11:52
WuestenfuxWuestenfux
4,4981413
$endgroup$
The tensor product of two $ntimes n$ matrices $A,B$ is
$$Aotimes B = left(begin{array}{ccc} a_{11}B & ldots& a_{1n}B\
vdots & ldots & vdots\
a_{n1}B & ldots & a_{nn} B
end{array}right).$$
So if $A,B$ are column stochastic, then so is $Aotimes B$. Just check the $j$-th column of the block matrix $Aotimes B$ and take inside the block matrix the $k$-th column of $B$:
$$a_{1j}[b_{1k}+ldots+b_{nk}] + ldots + a_{nj}[b_{1k}+ldots+b_{nk}] = a_{1j}+ldots+a_{nj} = 1.$$
Similar for the rows.
answered Nov 30 '18 at 11:52
WuestenfuxWuestenfux
4,4981413
answered Nov 30 '18 at 11:52
WuestenfuxWuestenfux
4,4981413
answered Nov 30 '18 at 11:52
WuestenfuxWuestenfux
4,4981413
answered Nov 30 '18 at 11:52
WuestenfuxWuestenfux
4,4981413
4,4981413
add a comment |
add a comment |
