Birkhoff representation of a stochastic matrix
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From the Birkhoff theorem, it is known that every doubly stochastic matrix can be written as a convex combination of permutation matrices, although this representation might not be unique.
Assume that a stochastic matrix is given. How can I find a permutation matrix that has a nonzero weight in at least one convex representation of the given stochastic matrix?
For example, if
$$P = beginbmatrix frac 12 & frac 12\ frac 12 & frac 12endbmatrix$$
then $P$ can be written as follows
$$P = frac 12 beginbmatrix 1 & 0\ 0 & 1endbmatrix + frac 12 beginbmatrix 0 & 1\ 1 & 0endbmatrix$$
In this case, both permutation matrices in the right-hand side have the property that I wish because they receive nonzero weight in at least one convex representation of $P$.
Is there any simple algorithm to rapidly find at least one such permutation matrix for a given stochastic matrix?
linear-algebra matrices permutations convex-hulls stochastic-matrices
edited Aug 17 at 7:31
Rodrigo de Azevedo
12.6k41751
asked Feb 24 '15 at 19:02
Saeid Haghighatshoar
311
add a comment |Â
up vote
6
down vote
favorite
From the Birkhoff theorem, it is known that every doubly stochastic matrix can be written as a convex combination of permutation matrices, although this representation might not be unique.
Assume that a stochastic matrix is given. How can I find a permutation matrix that has a nonzero weight in at least one convex representation of the given stochastic matrix?
For example, if
$$P = beginbmatrix frac 12 & frac 12\ frac 12 & frac 12endbmatrix$$
then $P$ can be written as follows
$$P = frac 12 beginbmatrix 1 & 0\ 0 & 1endbmatrix + frac 12 beginbmatrix 0 & 1\ 1 & 0endbmatrix$$
In this case, both permutation matrices in the right-hand side have the property that I wish because they receive nonzero weight in at least one convex representation of $P$.
Is there any simple algorithm to rapidly find at least one such permutation matrix for a given stochastic matrix?
linear-algebra matrices permutations convex-hulls stochastic-matrices
edited Aug 17 at 7:31
Rodrigo de Azevedo
12.6k41751
asked Feb 24 '15 at 19:02
Saeid Haghighatshoar
311
I thought of rewriting it as $P=(P-lambda I)+lambda I$ where $0<lambda<1$. But now, the sums of rows and columns in $(P-lambda I)$ is $1-lambda$ ?
– Srinivas K
Feb 24 '15 at 19:57
Thanks a lot. Yes, but the problem is that $P-lambda I$ might not be the convex combination of other permutation matrices.
– Saeid Haghighatshoar
Feb 25 '15 at 6:56
1
If $Q$ is a permutation matrix and $0<a<1$ such that all entries of $P-aQ$ are non-negative then $P=(1-a)fracP-aQ1-a+aQ$. Notice that $fracP-aQ1-a$ is doubly stochastic. Thus, a convex combination of permutations. Notice that $Q$'s weight is $a>0$. The converse is also true.
– Daniel
Aug 21 at 0:02
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
From the Birkhoff theorem, it is known that every doubly stochastic matrix can be written as a convex combination of permutation matrices, although this representation might not be unique.
Assume that a stochastic matrix is given. How can I find a permutation matrix that has a nonzero weight in at least one convex representation of the given stochastic matrix?
For example, if
$$P = beginbmatrix frac 12 & frac 12\ frac 12 & frac 12endbmatrix$$
then $P$ can be written as follows
$$P = frac 12 beginbmatrix 1 & 0\ 0 & 1endbmatrix + frac 12 beginbmatrix 0 & 1\ 1 & 0endbmatrix$$
In this case, both permutation matrices in the right-hand side have the property that I wish because they receive nonzero weight in at least one convex representation of $P$.
Is there any simple algorithm to rapidly find at least one such permutation matrix for a given stochastic matrix?
linear-algebra matrices permutations convex-hulls stochastic-matrices
edited Aug 17 at 7:31
Rodrigo de Azevedo
12.6k41751
asked Feb 24 '15 at 19:02
Saeid Haghighatshoar
311
From the Birkhoff theorem, it is known that every doubly stochastic matrix can be written as a convex combination of permutation matrices, although this representation might not be unique.
Assume that a stochastic matrix is given. How can I find a permutation matrix that has a nonzero weight in at least one convex representation of the given stochastic matrix?
For example, if
$$P = beginbmatrix frac 12 & frac 12\ frac 12 & frac 12endbmatrix$$
then $P$ can be written as follows
$$P = frac 12 beginbmatrix 1 & 0\ 0 & 1endbmatrix + frac 12 beginbmatrix 0 & 1\ 1 & 0endbmatrix$$
In this case, both permutation matrices in the right-hand side have the property that I wish because they receive nonzero weight in at least one convex representation of $P$.
Is there any simple algorithm to rapidly find at least one such permutation matrix for a given stochastic matrix?
linear-algebra matrices permutations convex-hulls stochastic-matrices
edited Aug 17 at 7:31
Rodrigo de Azevedo
12.6k41751
asked Feb 24 '15 at 19:02
Saeid Haghighatshoar
311
edited Aug 17 at 7:31
Rodrigo de Azevedo
12.6k41751
edited Aug 17 at 7:31
Rodrigo de Azevedo
12.6k41751
edited Aug 17 at 7:31
Rodrigo de Azevedo
12.6k41751
12.6k41751
asked Feb 24 '15 at 19:02
Saeid Haghighatshoar
311
asked Feb 24 '15 at 19:02
Saeid Haghighatshoar
311
asked Feb 24 '15 at 19:02
Saeid Haghighatshoar
311
311
I thought of rewriting it as $P=(P-lambda I)+lambda I$ where $0<lambda<1$. But now, the sums of rows and columns in $(P-lambda I)$ is $1-lambda$ ?
– Srinivas K
Feb 24 '15 at 19:57
Thanks a lot. Yes, but the problem is that $P-lambda I$ might not be the convex combination of other permutation matrices.
– Saeid Haghighatshoar
Feb 25 '15 at 6:56
1
If $Q$ is a permutation matrix and $0<a<1$ such that all entries of $P-aQ$ are non-negative then $P=(1-a)fracP-aQ1-a+aQ$. Notice that $fracP-aQ1-a$ is doubly stochastic. Thus, a convex combination of permutations. Notice that $Q$'s weight is $a>0$. The converse is also true.
– Daniel
Aug 21 at 0:02
add a comment |Â
I thought of rewriting it as $P=(P-lambda I)+lambda I$ where $0<lambda<1$. But now, the sums of rows and columns in $(P-lambda I)$ is $1-lambda$ ?
– Srinivas K
Feb 24 '15 at 19:57
Thanks a lot. Yes, but the problem is that $P-lambda I$ might not be the convex combination of other permutation matrices.
– Saeid Haghighatshoar
Feb 25 '15 at 6:56
1
If $Q$ is a permutation matrix and $0<a<1$ such that all entries of $P-aQ$ are non-negative then $P=(1-a)fracP-aQ1-a+aQ$. Notice that $fracP-aQ1-a$ is doubly stochastic. Thus, a convex combination of permutations. Notice that $Q$'s weight is $a>0$. The converse is also true.
– Daniel
Aug 21 at 0:02
I thought of rewriting it as $P=(P-lambda I)+lambda I$ where $0<lambda<1$. But now, the sums of rows and columns in $(P-lambda I)$ is $1-lambda$ ?
– Srinivas K
Feb 24 '15 at 19:57
I thought of rewriting it as $P=(P-lambda I)+lambda I$ where $0<lambda<1$. But now, the sums of rows and columns in $(P-lambda I)$ is $1-lambda$ ?
– Srinivas K
Feb 24 '15 at 19:57
Thanks a lot. Yes, but the problem is that $P-lambda I$ might not be the convex combination of other permutation matrices.
– Saeid Haghighatshoar
Feb 25 '15 at 6:56
Thanks a lot. Yes, but the problem is that $P-lambda I$ might not be the convex combination of other permutation matrices.
– Saeid Haghighatshoar
Feb 25 '15 at 6:56
1
1
If $Q$ is a permutation matrix and $0<a<1$ such that all entries of $P-aQ$ are non-negative then $P=(1-a)fracP-aQ1-a+aQ$. Notice that $fracP-aQ1-a$ is doubly stochastic. Thus, a convex combination of permutations. Notice that $Q$'s weight is $a>0$. The converse is also true.
– Daniel
Aug 21 at 0:02
If $Q$ is a permutation matrix and $0<a<1$ such that all entries of $P-aQ$ are non-negative then $P=(1-a)fracP-aQ1-a+aQ$. Notice that $fracP-aQ1-a$ is doubly stochastic. Thus, a convex combination of permutations. Notice that $Q$'s weight is $a>0$. The converse is also true.
– Daniel
Aug 21 at 0:02
add a comment |Â
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I thought of rewriting it as $P=(P-lambda I)+lambda I$ where $0<lambda<1$. But now, the sums of rows and columns in $(P-lambda I)$ is $1-lambda$ ?
– Srinivas K
Feb 24 '15 at 19:57
Thanks a lot. Yes, but the problem is that $P-lambda I$ might not be the convex combination of other permutation matrices.
– Saeid Haghighatshoar
Feb 25 '15 at 6:56
1
If $Q$ is a permutation matrix and $0<a<1$ such that all entries of $P-aQ$ are non-negative then $P=(1-a)fracP-aQ1-a+aQ$. Notice that $fracP-aQ1-a$ is doubly stochastic. Thus, a convex combination of permutations. Notice that $Q$'s weight is $a>0$. The converse is also true.
– Daniel
Aug 21 at 0:02