What can we say about invertible matrix $P$ under a condition that $A=PAP^-1$?

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Given an integral matrix $A$, we shall consider the set $$PAP^-1=A.$$
We can check that the set is a group under the matrix multiplication.



Can we say something algebraic for $P$? I am considering the concept of commutator to find some property for $P$. If you have another viewpoint or some comment, can you give it to me?




linear-algebra abstract-algebra inverse




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  • 6




    You can say that $A$ and $P$ commute.
    – Bernard
    Jul 26 at 15:58










  • What do you mean by an "integral matrix"?
    – xarles
    Jul 28 at 20:38










  • I meant a matrix whose entries are integers!
    – kswim
    Jul 29 at 16:12














up vote
5
down vote

favorite
1












Given an integral matrix $A$, we shall consider the set $$PAP^-1=A.$$
We can check that the set is a group under the matrix multiplication.



Can we say something algebraic for $P$? I am considering the concept of commutator to find some property for $P$. If you have another viewpoint or some comment, can you give it to me?




linear-algebra abstract-algebra inverse




share|cite|improve this question

















  • 6




    You can say that $A$ and $P$ commute.
    – Bernard
    Jul 26 at 15:58










  • What do you mean by an "integral matrix"?
    – xarles
    Jul 28 at 20:38










  • I meant a matrix whose entries are integers!
    – kswim
    Jul 29 at 16:12












up vote
5
down vote

favorite
1









up vote
5
down vote

favorite
1






1





Given an integral matrix $A$, we shall consider the set $$PAP^-1=A.$$
We can check that the set is a group under the matrix multiplication.



Can we say something algebraic for $P$? I am considering the concept of commutator to find some property for $P$. If you have another viewpoint or some comment, can you give it to me?




linear-algebra abstract-algebra inverse




share|cite|improve this question













Given an integral matrix $A$, we shall consider the set $$PAP^-1=A.$$
We can check that the set is a group under the matrix multiplication.



Can we say something algebraic for $P$? I am considering the concept of commutator to find some property for $P$. If you have another viewpoint or some comment, can you give it to me?





linear-algebra abstract-algebra inverse





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 27 at 5:02









user26857

38.7k123678




38.7k123678









asked Jul 26 at 15:49









kswim

1427




1427







  • 6




    You can say that $A$ and $P$ commute.
    – Bernard
    Jul 26 at 15:58










  • What do you mean by an "integral matrix"?
    – xarles
    Jul 28 at 20:38










  • I meant a matrix whose entries are integers!
    – kswim
    Jul 29 at 16:12












  • 6




    You can say that $A$ and $P$ commute.
    – Bernard
    Jul 26 at 15:58










  • What do you mean by an "integral matrix"?
    – xarles
    Jul 28 at 20:38










  • I meant a matrix whose entries are integers!
    – kswim
    Jul 29 at 16:12







6




6




You can say that $A$ and $P$ commute.
– Bernard
Jul 26 at 15:58




You can say that $A$ and $P$ commute.
– Bernard
Jul 26 at 15:58












What do you mean by an "integral matrix"?
– xarles
Jul 28 at 20:38




What do you mean by an "integral matrix"?
– xarles
Jul 28 at 20:38












I meant a matrix whose entries are integers!
– kswim
Jul 29 at 16:12




I meant a matrix whose entries are integers!
– kswim
Jul 29 at 16:12










1 Answer
1






active

oldest

votes

















up vote
1
down vote













Since you ask for something algebraic, the following is some more information that just commuting.



Let $mathbbF$ be a field. Denote by $C_A = AB=BA$ the centralizer of a matrix $A$. See also a neat way by Amritanshu Prasad to find the vector space dimension of $C_A$ over $mathbbF$. Denote by $G_A= AP=PA$. Then $G_A=Pin C_A$.



If we treat $mathbbF^n$ as a $mathbbF[x]$-module $M^A$ given by $xcdot v=Av$, then by the structure theorem for finitely generated modules over PID, we may write
$$
M^Asimeq bigopluslimits_p mathopopluslimits_i mathbbF[x]/(p^lambda_p,i),
$$
where $p$ runs over irreducible factor of the characteristic polynomial of $A$ and $lambda_p,i>0$ is the powers of $p$ appearing on the $p$-primary part of $M^A$. Then the centralizer of $A$ can be written as an $mathbbF[x]$-module endomorphism algebra
$
mathrmEnd_mathbbF[x](M^A).
$



Then we need invertible matrices in $
mathrmEnd_mathbbF[x](M^A).
$



In case when $mathbbF$ is a finite field, a method of obtaining such matrices $Pin G_A$ is outlined in Theorem 1.10.7 of Enumerative Combinatorics I. This method general and also applies when $mathbbF$ is not a finite field.






share|cite|improve this answer





















  • Do you have another literature in your mind about the centralizer of a matrix over an infinite field?
    – kswim
    Aug 3 at 18:50










  • I do not have another literature in my mind. For the centralizer of a matrix over an infinite field, the process of finding basis vectors work the same as in the finite field case.
    – i707107
    Aug 3 at 20:06










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote













Since you ask for something algebraic, the following is some more information that just commuting.



Let $mathbbF$ be a field. Denote by $C_A = AB=BA$ the centralizer of a matrix $A$. See also a neat way by Amritanshu Prasad to find the vector space dimension of $C_A$ over $mathbbF$. Denote by $G_A= AP=PA$. Then $G_A=Pin C_A$.



If we treat $mathbbF^n$ as a $mathbbF[x]$-module $M^A$ given by $xcdot v=Av$, then by the structure theorem for finitely generated modules over PID, we may write
$$
M^Asimeq bigopluslimits_p mathopopluslimits_i mathbbF[x]/(p^lambda_p,i),
$$
where $p$ runs over irreducible factor of the characteristic polynomial of $A$ and $lambda_p,i>0$ is the powers of $p$ appearing on the $p$-primary part of $M^A$. Then the centralizer of $A$ can be written as an $mathbbF[x]$-module endomorphism algebra
$
mathrmEnd_mathbbF[x](M^A).
$



Then we need invertible matrices in $
mathrmEnd_mathbbF[x](M^A).
$



In case when $mathbbF$ is a finite field, a method of obtaining such matrices $Pin G_A$ is outlined in Theorem 1.10.7 of Enumerative Combinatorics I. This method general and also applies when $mathbbF$ is not a finite field.






share|cite|improve this answer





















  • Do you have another literature in your mind about the centralizer of a matrix over an infinite field?
    – kswim
    Aug 3 at 18:50










  • I do not have another literature in my mind. For the centralizer of a matrix over an infinite field, the process of finding basis vectors work the same as in the finite field case.
    – i707107
    Aug 3 at 20:06














up vote
1
down vote













Since you ask for something algebraic, the following is some more information that just commuting.



Let $mathbbF$ be a field. Denote by $C_A = AB=BA$ the centralizer of a matrix $A$. See also a neat way by Amritanshu Prasad to find the vector space dimension of $C_A$ over $mathbbF$. Denote by $G_A= AP=PA$. Then $G_A=Pin C_A$.



If we treat $mathbbF^n$ as a $mathbbF[x]$-module $M^A$ given by $xcdot v=Av$, then by the structure theorem for finitely generated modules over PID, we may write
$$
M^Asimeq bigopluslimits_p mathopopluslimits_i mathbbF[x]/(p^lambda_p,i),
$$
where $p$ runs over irreducible factor of the characteristic polynomial of $A$ and $lambda_p,i>0$ is the powers of $p$ appearing on the $p$-primary part of $M^A$. Then the centralizer of $A$ can be written as an $mathbbF[x]$-module endomorphism algebra
$
mathrmEnd_mathbbF[x](M^A).
$



Then we need invertible matrices in $
mathrmEnd_mathbbF[x](M^A).
$



In case when $mathbbF$ is a finite field, a method of obtaining such matrices $Pin G_A$ is outlined in Theorem 1.10.7 of Enumerative Combinatorics I. This method general and also applies when $mathbbF$ is not a finite field.






share|cite|improve this answer





















  • Do you have another literature in your mind about the centralizer of a matrix over an infinite field?
    – kswim
    Aug 3 at 18:50










  • I do not have another literature in my mind. For the centralizer of a matrix over an infinite field, the process of finding basis vectors work the same as in the finite field case.
    – i707107
    Aug 3 at 20:06












up vote
1
down vote










up vote
1
down vote









Since you ask for something algebraic, the following is some more information that just commuting.



Let $mathbbF$ be a field. Denote by $C_A = AB=BA$ the centralizer of a matrix $A$. See also a neat way by Amritanshu Prasad to find the vector space dimension of $C_A$ over $mathbbF$. Denote by $G_A= AP=PA$. Then $G_A=Pin C_A$.



If we treat $mathbbF^n$ as a $mathbbF[x]$-module $M^A$ given by $xcdot v=Av$, then by the structure theorem for finitely generated modules over PID, we may write
$$
M^Asimeq bigopluslimits_p mathopopluslimits_i mathbbF[x]/(p^lambda_p,i),
$$
where $p$ runs over irreducible factor of the characteristic polynomial of $A$ and $lambda_p,i>0$ is the powers of $p$ appearing on the $p$-primary part of $M^A$. Then the centralizer of $A$ can be written as an $mathbbF[x]$-module endomorphism algebra
$
mathrmEnd_mathbbF[x](M^A).
$



Then we need invertible matrices in $
mathrmEnd_mathbbF[x](M^A).
$



In case when $mathbbF$ is a finite field, a method of obtaining such matrices $Pin G_A$ is outlined in Theorem 1.10.7 of Enumerative Combinatorics I. This method general and also applies when $mathbbF$ is not a finite field.






share|cite|improve this answer













Since you ask for something algebraic, the following is some more information that just commuting.



Let $mathbbF$ be a field. Denote by $C_A = AB=BA$ the centralizer of a matrix $A$. See also a neat way by Amritanshu Prasad to find the vector space dimension of $C_A$ over $mathbbF$. Denote by $G_A= AP=PA$. Then $G_A=Pin C_A$.



If we treat $mathbbF^n$ as a $mathbbF[x]$-module $M^A$ given by $xcdot v=Av$, then by the structure theorem for finitely generated modules over PID, we may write
$$
M^Asimeq bigopluslimits_p mathopopluslimits_i mathbbF[x]/(p^lambda_p,i),
$$
where $p$ runs over irreducible factor of the characteristic polynomial of $A$ and $lambda_p,i>0$ is the powers of $p$ appearing on the $p$-primary part of $M^A$. Then the centralizer of $A$ can be written as an $mathbbF[x]$-module endomorphism algebra
$
mathrmEnd_mathbbF[x](M^A).
$



Then we need invertible matrices in $
mathrmEnd_mathbbF[x](M^A).
$



In case when $mathbbF$ is a finite field, a method of obtaining such matrices $Pin G_A$ is outlined in Theorem 1.10.7 of Enumerative Combinatorics I. This method general and also applies when $mathbbF$ is not a finite field.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 28 at 15:39









i707107

11.2k21241




11.2k21241











  • Do you have another literature in your mind about the centralizer of a matrix over an infinite field?
    – kswim
    Aug 3 at 18:50










  • I do not have another literature in my mind. For the centralizer of a matrix over an infinite field, the process of finding basis vectors work the same as in the finite field case.
    – i707107
    Aug 3 at 20:06
















  • Do you have another literature in your mind about the centralizer of a matrix over an infinite field?
    – kswim
    Aug 3 at 18:50










  • I do not have another literature in my mind. For the centralizer of a matrix over an infinite field, the process of finding basis vectors work the same as in the finite field case.
    – i707107
    Aug 3 at 20:06















Do you have another literature in your mind about the centralizer of a matrix over an infinite field?
– kswim
Aug 3 at 18:50




Do you have another literature in your mind about the centralizer of a matrix over an infinite field?
– kswim
Aug 3 at 18:50












I do not have another literature in my mind. For the centralizer of a matrix over an infinite field, the process of finding basis vectors work the same as in the finite field case.
– i707107
Aug 3 at 20:06




I do not have another literature in my mind. For the centralizer of a matrix over an infinite field, the process of finding basis vectors work the same as in the finite field case.
– i707107
Aug 3 at 20:06












 

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