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What can we say about invertible matrix $P$ under a condition that $A=PAP^-1$?
Clash Royale CLAN TAG#URR8PPP
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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
edited Jul 27 at 5:02
user26857
38.7k123678
asked Jul 26 at 15:49
kswim
1427
add a comment |Â
up vote
5
down vote
favorite
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
edited Jul 27 at 5:02
user26857
38.7k123678
asked Jul 26 at 15:49
kswim
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
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
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
edited Jul 27 at 5:02
user26857
38.7k123678
asked Jul 26 at 15:49
kswim
1427
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
edited Jul 27 at 5:02
user26857
38.7k123678
asked Jul 26 at 15:49
kswim
1427
edited Jul 27 at 5:02
user26857
38.7k123678
edited Jul 27 at 5:02
user26857
38.7k123678
edited Jul 27 at 5:02
user26857
38.7k123678
38.7k123678
asked Jul 26 at 15:49
kswim
1427
asked Jul 26 at 15:49
kswim
1427
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
add a comment |Â
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
add a comment |Â
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.
answered Jul 28 at 15:39
i707107
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
add a comment |Â
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.
answered Jul 28 at 15:39
i707107
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
add a comment |Â
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.
answered Jul 28 at 15:39
i707107
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
add a comment |Â
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.
answered Jul 28 at 15:39
i707107
11.2k21241
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.
answered Jul 28 at 15:39
i707107
11.2k21241
answered Jul 28 at 15:39
i707107
11.2k21241
answered Jul 28 at 15:39
i707107
11.2k21241
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
add a comment |Â
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
add a comment |Â
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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