Inverse of a regular stochastic matrix
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Is it true that the inverse of a regular stochastic matrix is also regular? Are there any other interesting features that the inverse may have of a regular stochastic matrix?
Hope someone could answer these questions. Thanks in advance.
matrices inverse stochastic-matrices
edited Aug 17 at 7:35
Rodrigo de Azevedo
12.6k41751
asked May 3 '14 at 14:09
Heisenberg
1,3271637
add a comment |Â
up vote
2
down vote
favorite
Is it true that the inverse of a regular stochastic matrix is also regular? Are there any other interesting features that the inverse may have of a regular stochastic matrix?
Hope someone could answer these questions. Thanks in advance.
matrices inverse stochastic-matrices
edited Aug 17 at 7:35
Rodrigo de Azevedo
12.6k41751
asked May 3 '14 at 14:09
Heisenberg
1,3271637
The inverse of any regular matrix is regular.
– MartÃn-Blas Pérez Pinilla
May 5 '14 at 9:50
It might be helpful to add the definitions of stochastic matrix and regular stochastic matrix to the question.
– Juho Kokkala
May 5 '14 at 10:19
@MartÃn-BlasPérezPinilla how about the counter example given by Juho below?
– Heisenberg
May 6 '14 at 5:29
@Heisenberg, I was thinking in regular=invertible. In Spanish is the most common meaning.
– MartÃn-Blas Pérez Pinilla
May 6 '14 at 6:12
In 2015, Reza Farhadian showed that there are some regular doubly stochastic matrices such that their inverses are regular doubly stochastic matrices [1]. Also, you can see [2, Appendix]. [1] neda.irstat.ir/article-1-229-fa.html. [2] Reza Farhadian, Nader Asadian F., On a New Class of Regular Doubly Stochastic Processes, American Journal of Theoretical and Applied Statistics. Vol. 6, No. 3, 2017, pp. 156-160. doi: 10.11648/j.ajtas.20170603.14
– user452330
Jun 14 '17 at 17:30
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Is it true that the inverse of a regular stochastic matrix is also regular? Are there any other interesting features that the inverse may have of a regular stochastic matrix?
Hope someone could answer these questions. Thanks in advance.
matrices inverse stochastic-matrices
edited Aug 17 at 7:35
Rodrigo de Azevedo
12.6k41751
asked May 3 '14 at 14:09
Heisenberg
1,3271637
Is it true that the inverse of a regular stochastic matrix is also regular? Are there any other interesting features that the inverse may have of a regular stochastic matrix?
Hope someone could answer these questions. Thanks in advance.
matrices inverse stochastic-matrices
edited Aug 17 at 7:35
Rodrigo de Azevedo
12.6k41751
asked May 3 '14 at 14:09
Heisenberg
1,3271637
edited Aug 17 at 7:35
Rodrigo de Azevedo
12.6k41751
edited Aug 17 at 7:35
Rodrigo de Azevedo
12.6k41751
edited Aug 17 at 7:35
Rodrigo de Azevedo
12.6k41751
12.6k41751
asked May 3 '14 at 14:09
Heisenberg
1,3271637
asked May 3 '14 at 14:09
Heisenberg
1,3271637
asked May 3 '14 at 14:09
Heisenberg
1,3271637
1,3271637
The inverse of any regular matrix is regular.
– MartÃn-Blas Pérez Pinilla
May 5 '14 at 9:50
It might be helpful to add the definitions of stochastic matrix and regular stochastic matrix to the question.
– Juho Kokkala
May 5 '14 at 10:19
@MartÃn-BlasPérezPinilla how about the counter example given by Juho below?
– Heisenberg
May 6 '14 at 5:29
@Heisenberg, I was thinking in regular=invertible. In Spanish is the most common meaning.
– MartÃn-Blas Pérez Pinilla
May 6 '14 at 6:12
In 2015, Reza Farhadian showed that there are some regular doubly stochastic matrices such that their inverses are regular doubly stochastic matrices [1]. Also, you can see [2, Appendix]. [1] neda.irstat.ir/article-1-229-fa.html. [2] Reza Farhadian, Nader Asadian F., On a New Class of Regular Doubly Stochastic Processes, American Journal of Theoretical and Applied Statistics. Vol. 6, No. 3, 2017, pp. 156-160. doi: 10.11648/j.ajtas.20170603.14
– user452330
Jun 14 '17 at 17:30
add a comment |Â
The inverse of any regular matrix is regular.
– MartÃn-Blas Pérez Pinilla
May 5 '14 at 9:50
It might be helpful to add the definitions of stochastic matrix and regular stochastic matrix to the question.
– Juho Kokkala
May 5 '14 at 10:19
@MartÃn-BlasPérezPinilla how about the counter example given by Juho below?
– Heisenberg
May 6 '14 at 5:29
@Heisenberg, I was thinking in regular=invertible. In Spanish is the most common meaning.
– MartÃn-Blas Pérez Pinilla
May 6 '14 at 6:12
In 2015, Reza Farhadian showed that there are some regular doubly stochastic matrices such that their inverses are regular doubly stochastic matrices [1]. Also, you can see [2, Appendix]. [1] neda.irstat.ir/article-1-229-fa.html. [2] Reza Farhadian, Nader Asadian F., On a New Class of Regular Doubly Stochastic Processes, American Journal of Theoretical and Applied Statistics. Vol. 6, No. 3, 2017, pp. 156-160. doi: 10.11648/j.ajtas.20170603.14
– user452330
Jun 14 '17 at 17:30
The inverse of any regular matrix is regular.
– MartÃn-Blas Pérez Pinilla
May 5 '14 at 9:50
The inverse of any regular matrix is regular.
– MartÃn-Blas Pérez Pinilla
May 5 '14 at 9:50
It might be helpful to add the definitions of stochastic matrix and regular stochastic matrix to the question.
– Juho Kokkala
May 5 '14 at 10:19
It might be helpful to add the definitions of stochastic matrix and regular stochastic matrix to the question.
– Juho Kokkala
May 5 '14 at 10:19
@MartÃn-BlasPérezPinilla how about the counter example given by Juho below?
– Heisenberg
May 6 '14 at 5:29
@MartÃn-BlasPérezPinilla how about the counter example given by Juho below?
– Heisenberg
May 6 '14 at 5:29
@Heisenberg, I was thinking in regular=invertible. In Spanish is the most common meaning.
– MartÃn-Blas Pérez Pinilla
May 6 '14 at 6:12
@Heisenberg, I was thinking in regular=invertible. In Spanish is the most common meaning.
– MartÃn-Blas Pérez Pinilla
May 6 '14 at 6:12
In 2015, Reza Farhadian showed that there are some regular doubly stochastic matrices such that their inverses are regular doubly stochastic matrices [1]. Also, you can see [2, Appendix]. [1] neda.irstat.ir/article-1-229-fa.html. [2] Reza Farhadian, Nader Asadian F., On a New Class of Regular Doubly Stochastic Processes, American Journal of Theoretical and Applied Statistics. Vol. 6, No. 3, 2017, pp. 156-160. doi: 10.11648/j.ajtas.20170603.14
– user452330
Jun 14 '17 at 17:30
In 2015, Reza Farhadian showed that there are some regular doubly stochastic matrices such that their inverses are regular doubly stochastic matrices [1]. Also, you can see [2, Appendix]. [1] neda.irstat.ir/article-1-229-fa.html. [2] Reza Farhadian, Nader Asadian F., On a New Class of Regular Doubly Stochastic Processes, American Journal of Theoretical and Applied Statistics. Vol. 6, No. 3, 2017, pp. 156-160. doi: 10.11648/j.ajtas.20170603.14
– user452330
Jun 14 '17 at 17:30
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
Counterexample: consider the following stochastic matrix:
beginequation
A = beginpmatrix 2/3 & 1/3 \ 1/3 & 2/3 endpmatrix.
endequation
Now, the inverse is
beginequation
A^-1 = beginpmatrix 2 & -1 \ -1 & 2 endpmatrix,
endequation
which is not even stochastic (entries are not nonnegative), let alone regular.
All powers of $A^-1$ also have negative nondiagonal entries, but this is not even needed as the definition of regular stochastic matrix requires stochasticity.
answered May 5 '14 at 10:37
Juho Kokkala
597210
Thanks a lot. So there is no reason to study inverse of regular stochastic matrices right?
– Heisenberg
May 6 '14 at 5:26
Note that the row sums of the inverse of a right stochastic matrix must equal 1, and the column sums of the inverse of a left stochastic matrix must equal 1, because of math.stackexchange.com/questions/946776/…
– Simon
Sep 25 '16 at 12:38
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Counterexample: consider the following stochastic matrix:
beginequation
A = beginpmatrix 2/3 & 1/3 \ 1/3 & 2/3 endpmatrix.
endequation
Now, the inverse is
beginequation
A^-1 = beginpmatrix 2 & -1 \ -1 & 2 endpmatrix,
endequation
which is not even stochastic (entries are not nonnegative), let alone regular.
All powers of $A^-1$ also have negative nondiagonal entries, but this is not even needed as the definition of regular stochastic matrix requires stochasticity.
answered May 5 '14 at 10:37
Juho Kokkala
597210
Thanks a lot. So there is no reason to study inverse of regular stochastic matrices right?
– Heisenberg
May 6 '14 at 5:26
Note that the row sums of the inverse of a right stochastic matrix must equal 1, and the column sums of the inverse of a left stochastic matrix must equal 1, because of math.stackexchange.com/questions/946776/…
– Simon
Sep 25 '16 at 12:38
add a comment |Â
up vote
2
down vote
accepted
Counterexample: consider the following stochastic matrix:
beginequation
A = beginpmatrix 2/3 & 1/3 \ 1/3 & 2/3 endpmatrix.
endequation
Now, the inverse is
beginequation
A^-1 = beginpmatrix 2 & -1 \ -1 & 2 endpmatrix,
endequation
which is not even stochastic (entries are not nonnegative), let alone regular.
All powers of $A^-1$ also have negative nondiagonal entries, but this is not even needed as the definition of regular stochastic matrix requires stochasticity.
answered May 5 '14 at 10:37
Juho Kokkala
597210
Thanks a lot. So there is no reason to study inverse of regular stochastic matrices right?
– Heisenberg
May 6 '14 at 5:26
Note that the row sums of the inverse of a right stochastic matrix must equal 1, and the column sums of the inverse of a left stochastic matrix must equal 1, because of math.stackexchange.com/questions/946776/…
– Simon
Sep 25 '16 at 12:38
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Counterexample: consider the following stochastic matrix:
beginequation
A = beginpmatrix 2/3 & 1/3 \ 1/3 & 2/3 endpmatrix.
endequation
Now, the inverse is
beginequation
A^-1 = beginpmatrix 2 & -1 \ -1 & 2 endpmatrix,
endequation
which is not even stochastic (entries are not nonnegative), let alone regular.
All powers of $A^-1$ also have negative nondiagonal entries, but this is not even needed as the definition of regular stochastic matrix requires stochasticity.
answered May 5 '14 at 10:37
Juho Kokkala
597210
Counterexample: consider the following stochastic matrix:
beginequation
A = beginpmatrix 2/3 & 1/3 \ 1/3 & 2/3 endpmatrix.
endequation
Now, the inverse is
beginequation
A^-1 = beginpmatrix 2 & -1 \ -1 & 2 endpmatrix,
endequation
which is not even stochastic (entries are not nonnegative), let alone regular.
All powers of $A^-1$ also have negative nondiagonal entries, but this is not even needed as the definition of regular stochastic matrix requires stochasticity.
answered May 5 '14 at 10:37
Juho Kokkala
597210
answered May 5 '14 at 10:37
Juho Kokkala
597210
answered May 5 '14 at 10:37
Juho Kokkala
597210
answered May 5 '14 at 10:37
Juho Kokkala
597210
597210
Thanks a lot. So there is no reason to study inverse of regular stochastic matrices right?
– Heisenberg
May 6 '14 at 5:26
Note that the row sums of the inverse of a right stochastic matrix must equal 1, and the column sums of the inverse of a left stochastic matrix must equal 1, because of math.stackexchange.com/questions/946776/…
– Simon
Sep 25 '16 at 12:38
add a comment |Â
Thanks a lot. So there is no reason to study inverse of regular stochastic matrices right?
– Heisenberg
May 6 '14 at 5:26
Note that the row sums of the inverse of a right stochastic matrix must equal 1, and the column sums of the inverse of a left stochastic matrix must equal 1, because of math.stackexchange.com/questions/946776/…
– Simon
Sep 25 '16 at 12:38
Thanks a lot. So there is no reason to study inverse of regular stochastic matrices right?
– Heisenberg
May 6 '14 at 5:26
Thanks a lot. So there is no reason to study inverse of regular stochastic matrices right?
– Heisenberg
May 6 '14 at 5:26
Note that the row sums of the inverse of a right stochastic matrix must equal 1, and the column sums of the inverse of a left stochastic matrix must equal 1, because of math.stackexchange.com/questions/946776/…
– Simon
Sep 25 '16 at 12:38
Note that the row sums of the inverse of a right stochastic matrix must equal 1, and the column sums of the inverse of a left stochastic matrix must equal 1, because of math.stackexchange.com/questions/946776/…
– Simon
Sep 25 '16 at 12:38
add a comment |Â
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The inverse of any regular matrix is regular.
– MartÃn-Blas Pérez Pinilla
May 5 '14 at 9:50
It might be helpful to add the definitions of stochastic matrix and regular stochastic matrix to the question.
– Juho Kokkala
May 5 '14 at 10:19
@MartÃn-BlasPérezPinilla how about the counter example given by Juho below?
– Heisenberg
May 6 '14 at 5:29
@Heisenberg, I was thinking in regular=invertible. In Spanish is the most common meaning.
– MartÃn-Blas Pérez Pinilla
May 6 '14 at 6:12
In 2015, Reza Farhadian showed that there are some regular doubly stochastic matrices such that their inverses are regular doubly stochastic matrices [1]. Also, you can see [2, Appendix]. [1] neda.irstat.ir/article-1-229-fa.html. [2] Reza Farhadian, Nader Asadian F., On a New Class of Regular Doubly Stochastic Processes, American Journal of Theoretical and Applied Statistics. Vol. 6, No. 3, 2017, pp. 156-160. doi: 10.11648/j.ajtas.20170603.14
– user452330
Jun 14 '17 at 17:30