Orthogonality of stochastic matrix
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Given a column stochastic matrix $P$, I wanted to give a relation between $|P|$ and orthogonality of $P$.
One simple way to think about how close $P$ is to being orthogonal is $|P^topP - I|$. Then, I simply went ahead and wrote:
$$|P| leq sqrt$$
First of all, does the above make sense? Secondly, I am not utilizing the fact that $P$ is stochastic, are there ways to get any better relationship than the one given above? Are there any measures defined for stochastic matrices (in literature) that measure how close a matrix is to being orthogonal?
P.S. All norms are Frobenius norms.
matrices orthogonal-matrices stochastic-matrices
edited Aug 17 at 7:14
Rodrigo de Azevedo
12.6k41751
asked Oct 26 '15 at 14:26
TenaliRaman
3,0891222
add a comment |Â
up vote
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down vote
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Given a column stochastic matrix $P$, I wanted to give a relation between $|P|$ and orthogonality of $P$.
One simple way to think about how close $P$ is to being orthogonal is $|P^topP - I|$. Then, I simply went ahead and wrote:
$$|P| leq sqrt$$
First of all, does the above make sense? Secondly, I am not utilizing the fact that $P$ is stochastic, are there ways to get any better relationship than the one given above? Are there any measures defined for stochastic matrices (in literature) that measure how close a matrix is to being orthogonal?
P.S. All norms are Frobenius norms.
matrices orthogonal-matrices stochastic-matrices
edited Aug 17 at 7:14
Rodrigo de Azevedo
12.6k41751
asked Oct 26 '15 at 14:26
TenaliRaman
3,0891222
1
For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
– Algebraic Pavel
Oct 26 '15 at 16:28
@AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
– TenaliRaman
Oct 26 '15 at 17:27
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given a column stochastic matrix $P$, I wanted to give a relation between $|P|$ and orthogonality of $P$.
One simple way to think about how close $P$ is to being orthogonal is $|P^topP - I|$. Then, I simply went ahead and wrote:
$$|P| leq sqrt$$
First of all, does the above make sense? Secondly, I am not utilizing the fact that $P$ is stochastic, are there ways to get any better relationship than the one given above? Are there any measures defined for stochastic matrices (in literature) that measure how close a matrix is to being orthogonal?
P.S. All norms are Frobenius norms.
matrices orthogonal-matrices stochastic-matrices
edited Aug 17 at 7:14
Rodrigo de Azevedo
12.6k41751
asked Oct 26 '15 at 14:26
TenaliRaman
3,0891222
Given a column stochastic matrix $P$, I wanted to give a relation between $|P|$ and orthogonality of $P$.
One simple way to think about how close $P$ is to being orthogonal is $|P^topP - I|$. Then, I simply went ahead and wrote:
$$|P| leq sqrt$$
First of all, does the above make sense? Secondly, I am not utilizing the fact that $P$ is stochastic, are there ways to get any better relationship than the one given above? Are there any measures defined for stochastic matrices (in literature) that measure how close a matrix is to being orthogonal?
P.S. All norms are Frobenius norms.
matrices orthogonal-matrices stochastic-matrices
edited Aug 17 at 7:14
Rodrigo de Azevedo
12.6k41751
asked Oct 26 '15 at 14:26
TenaliRaman
3,0891222
edited Aug 17 at 7:14
Rodrigo de Azevedo
12.6k41751
edited Aug 17 at 7:14
Rodrigo de Azevedo
12.6k41751
edited Aug 17 at 7:14
Rodrigo de Azevedo
12.6k41751
12.6k41751
asked Oct 26 '15 at 14:26
TenaliRaman
3,0891222
asked Oct 26 '15 at 14:26
TenaliRaman
3,0891222
asked Oct 26 '15 at 14:26
TenaliRaman
3,0891222
3,0891222
1
For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
– Algebraic Pavel
Oct 26 '15 at 16:28
@AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
– TenaliRaman
Oct 26 '15 at 17:27
add a comment |Â
1
For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
– Algebraic Pavel
Oct 26 '15 at 16:28
@AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
– TenaliRaman
Oct 26 '15 at 17:27
1
1
For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
– Algebraic Pavel
Oct 26 '15 at 16:28
For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
– Algebraic Pavel
Oct 26 '15 at 16:28
@AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
– TenaliRaman
Oct 26 '15 at 17:27
@AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
– TenaliRaman
Oct 26 '15 at 17:27
add a comment |Â
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1
For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
– Algebraic Pavel
Oct 26 '15 at 16:28
@AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
– TenaliRaman
Oct 26 '15 at 17:27