Stochastic matrix question
Clash Royale CLAN TAG#URR8PPP
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A stochastic matrix is one which each column sum equal one.
$$P= beginbmatrix P_11 & P_12 & ldots & P_1N \
P_21 & P_22 & ldots & P_2N \
ldots & ldots & ldots & ldots \
P_N1 & P_N2 & ldots & P_NN endbmatrix$$
Show that is $P$ is a stochastic matrix. Then $P^2$ is a stochastic matrix.
Then show $P^n$ is stochastic for all postive integer N.
I am not sure how to do this I think by induction. I know stochastic is that $forall i $ $sum_j=1^Np_ij=1$
probability matrices stochastic-matrices
edited Aug 17 at 7:09
Rodrigo de Azevedo
12.6k41751
asked Feb 2 at 20:02
Fernando Martinez
3,29784076
add a comment |Â
up vote
0
down vote
favorite
A stochastic matrix is one which each column sum equal one.
$$P= beginbmatrix P_11 & P_12 & ldots & P_1N \
P_21 & P_22 & ldots & P_2N \
ldots & ldots & ldots & ldots \
P_N1 & P_N2 & ldots & P_NN endbmatrix$$
Show that is $P$ is a stochastic matrix. Then $P^2$ is a stochastic matrix.
Then show $P^n$ is stochastic for all postive integer N.
I am not sure how to do this I think by induction. I know stochastic is that $forall i $ $sum_j=1^Np_ij=1$
probability matrices stochastic-matrices
edited Aug 17 at 7:09
Rodrigo de Azevedo
12.6k41751
asked Feb 2 at 20:02
Fernando Martinez
3,29784076
Can you do the first part and show that the each column of $P^2$ sums to one?
– Harto Saarinen
Feb 2 at 20:05
Usually a stochastic matrix is one for which the row sums are one.
– carmichael561
Feb 2 at 20:07
Actually $sum_j=1^Np_ij=1$ says each row has sum $1$.
– David C. Ullrich
Feb 2 at 20:09
I messed up in my book it says each column sum equal one
– Fernando Martinez
Feb 2 at 20:10
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A stochastic matrix is one which each column sum equal one.
$$P= beginbmatrix P_11 & P_12 & ldots & P_1N \
P_21 & P_22 & ldots & P_2N \
ldots & ldots & ldots & ldots \
P_N1 & P_N2 & ldots & P_NN endbmatrix$$
Show that is $P$ is a stochastic matrix. Then $P^2$ is a stochastic matrix.
Then show $P^n$ is stochastic for all postive integer N.
I am not sure how to do this I think by induction. I know stochastic is that $forall i $ $sum_j=1^Np_ij=1$
probability matrices stochastic-matrices
edited Aug 17 at 7:09
Rodrigo de Azevedo
12.6k41751
asked Feb 2 at 20:02
Fernando Martinez
3,29784076
A stochastic matrix is one which each column sum equal one.
$$P= beginbmatrix P_11 & P_12 & ldots & P_1N \
P_21 & P_22 & ldots & P_2N \
ldots & ldots & ldots & ldots \
P_N1 & P_N2 & ldots & P_NN endbmatrix$$
Show that is $P$ is a stochastic matrix. Then $P^2$ is a stochastic matrix.
Then show $P^n$ is stochastic for all postive integer N.
I am not sure how to do this I think by induction. I know stochastic is that $forall i $ $sum_j=1^Np_ij=1$
probability matrices stochastic-matrices
edited Aug 17 at 7:09
Rodrigo de Azevedo
12.6k41751
asked Feb 2 at 20:02
Fernando Martinez
3,29784076
edited Aug 17 at 7:09
Rodrigo de Azevedo
12.6k41751
edited Aug 17 at 7:09
Rodrigo de Azevedo
12.6k41751
edited Aug 17 at 7:09
Rodrigo de Azevedo
12.6k41751
12.6k41751
asked Feb 2 at 20:02
Fernando Martinez
3,29784076
asked Feb 2 at 20:02
Fernando Martinez
3,29784076
asked Feb 2 at 20:02
Fernando Martinez
3,29784076
3,29784076
Can you do the first part and show that the each column of $P^2$ sums to one?
– Harto Saarinen
Feb 2 at 20:05
Usually a stochastic matrix is one for which the row sums are one.
– carmichael561
Feb 2 at 20:07
Actually $sum_j=1^Np_ij=1$ says each row has sum $1$.
– David C. Ullrich
Feb 2 at 20:09
I messed up in my book it says each column sum equal one
– Fernando Martinez
Feb 2 at 20:10
add a comment |Â
Can you do the first part and show that the each column of $P^2$ sums to one?
– Harto Saarinen
Feb 2 at 20:05
Usually a stochastic matrix is one for which the row sums are one.
– carmichael561
Feb 2 at 20:07
Actually $sum_j=1^Np_ij=1$ says each row has sum $1$.
– David C. Ullrich
Feb 2 at 20:09
I messed up in my book it says each column sum equal one
– Fernando Martinez
Feb 2 at 20:10
Can you do the first part and show that the each column of $P^2$ sums to one?
– Harto Saarinen
Feb 2 at 20:05
Can you do the first part and show that the each column of $P^2$ sums to one?
– Harto Saarinen
Feb 2 at 20:05
Usually a stochastic matrix is one for which the row sums are one.
– carmichael561
Feb 2 at 20:07
Usually a stochastic matrix is one for which the row sums are one.
– carmichael561
Feb 2 at 20:07
Actually $sum_j=1^Np_ij=1$ says each row has sum $1$.
– David C. Ullrich
Feb 2 at 20:09
Actually $sum_j=1^Np_ij=1$ says each row has sum $1$.
– David C. Ullrich
Feb 2 at 20:09
I messed up in my book it says each column sum equal one
– Fernando Martinez
Feb 2 at 20:10
I messed up in my book it says each column sum equal one
– Fernando Martinez
Feb 2 at 20:10
add a comment |Â
2 Answers
2
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oldest
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up vote
1
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Let $e$ be the all one-matrix, the condition for $P$ to be a stochastic matrix can be rewritten as $$e^TP=e^T.$$
Now, let me try to show that $P^2$ is a stochastic matrix.
$$e^TP^2=(e^TP)P=(e^T)P=e^TPe=e^T$$
Hence $P^2$ is stochastic.
I will leave the case for $P^n$ as an exercise.
answered Feb 2 at 20:09
Siong Thye Goh
79.7k135299
add a comment |Â
up vote
0
down vote
Check any column of $P^2$, it's linear combination of columns of $P$, where the coefficients are again the entries of columns of $P$. Hence the sum of the entries of any column of $P^2$ is 1.
answered Feb 2 at 20:26
Ranveer Singh
19919
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Let $e$ be the all one-matrix, the condition for $P$ to be a stochastic matrix can be rewritten as $$e^TP=e^T.$$
Now, let me try to show that $P^2$ is a stochastic matrix.
$$e^TP^2=(e^TP)P=(e^T)P=e^TPe=e^T$$
Hence $P^2$ is stochastic.
I will leave the case for $P^n$ as an exercise.
answered Feb 2 at 20:09
Siong Thye Goh
79.7k135299
add a comment |Â
up vote
1
down vote
Let $e$ be the all one-matrix, the condition for $P$ to be a stochastic matrix can be rewritten as $$e^TP=e^T.$$
Now, let me try to show that $P^2$ is a stochastic matrix.
$$e^TP^2=(e^TP)P=(e^T)P=e^TPe=e^T$$
Hence $P^2$ is stochastic.
I will leave the case for $P^n$ as an exercise.
answered Feb 2 at 20:09
Siong Thye Goh
79.7k135299
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Let $e$ be the all one-matrix, the condition for $P$ to be a stochastic matrix can be rewritten as $$e^TP=e^T.$$
Now, let me try to show that $P^2$ is a stochastic matrix.
$$e^TP^2=(e^TP)P=(e^T)P=e^TPe=e^T$$
Hence $P^2$ is stochastic.
I will leave the case for $P^n$ as an exercise.
answered Feb 2 at 20:09
Siong Thye Goh
79.7k135299
Let $e$ be the all one-matrix, the condition for $P$ to be a stochastic matrix can be rewritten as $$e^TP=e^T.$$
Now, let me try to show that $P^2$ is a stochastic matrix.
$$e^TP^2=(e^TP)P=(e^T)P=e^TPe=e^T$$
Hence $P^2$ is stochastic.
I will leave the case for $P^n$ as an exercise.
answered Feb 2 at 20:09
Siong Thye Goh
79.7k135299
answered Feb 2 at 20:09
Siong Thye Goh
79.7k135299
answered Feb 2 at 20:09
Siong Thye Goh
79.7k135299
answered Feb 2 at 20:09
Siong Thye Goh
79.7k135299
79.7k135299
add a comment |Â
add a comment |Â
up vote
0
down vote
Check any column of $P^2$, it's linear combination of columns of $P$, where the coefficients are again the entries of columns of $P$. Hence the sum of the entries of any column of $P^2$ is 1.
answered Feb 2 at 20:26
Ranveer Singh
19919
add a comment |Â
up vote
0
down vote
Check any column of $P^2$, it's linear combination of columns of $P$, where the coefficients are again the entries of columns of $P$. Hence the sum of the entries of any column of $P^2$ is 1.
answered Feb 2 at 20:26
Ranveer Singh
19919
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Check any column of $P^2$, it's linear combination of columns of $P$, where the coefficients are again the entries of columns of $P$. Hence the sum of the entries of any column of $P^2$ is 1.
answered Feb 2 at 20:26
Ranveer Singh
19919
Check any column of $P^2$, it's linear combination of columns of $P$, where the coefficients are again the entries of columns of $P$. Hence the sum of the entries of any column of $P^2$ is 1.
answered Feb 2 at 20:26
Ranveer Singh
19919
answered Feb 2 at 20:26
Ranveer Singh
19919
answered Feb 2 at 20:26
Ranveer Singh
19919
answered Feb 2 at 20:26
Ranveer Singh
19919
19919
add a comment |Â
add a comment |Â
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Can you do the first part and show that the each column of $P^2$ sums to one?
– Harto Saarinen
Feb 2 at 20:05
Usually a stochastic matrix is one for which the row sums are one.
– carmichael561
Feb 2 at 20:07
Actually $sum_j=1^Np_ij=1$ says each row has sum $1$.
– David C. Ullrich
Feb 2 at 20:09
I messed up in my book it says each column sum equal one
– Fernando Martinez
Feb 2 at 20:10