Conditions when a permutation matrix is symmetric












1












$begingroup$


I am now playing with permutation matrices, http://mathworld.wolfram.com/PermutationMatrix.html.



Also, there is a similar discussion: Symmetric Permutation Matrix.
I want to ask more details than this one.



As we know, a permutation matrix is orthogonal, i.e., $E^T=E^{-1}$. I am interested in when it is symmetric, i.e., $E^T=E^{-1} = E$



Suppose




  1. Start from an identity matrix $I_n$.


  2. $n$ can be even or odd number.

  3. Pick $(i,j)$, where $0<i,jleq n$ and $i, j$ are integer. Exchange $i$-th and $j$-th columns of $I_n$ (identity matrix) and get $E$. Then $E$ is symmetric. This is because $E_{ii}=E_{jj}=0$ and $E_{ij}=E_{ji}=1$.

  4. Based on 3., if I pick a number of sets $(i,j)$, $(k,l)$, $(q,r), ldots$, without repeated index in each $(cdot,cdot)$, and permute columns of $I_n$ according to these sets, then the resulting permutation matrix $E$ is symmetric.


One key thing here is "without repeated index in each $(cdot,cdot)$". This is because if I do $(1,2)$ and $(2,3)$ for $I_3$ for example, I get



$$begin{bmatrix}0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 end{bmatrix},$$



which is not symmetric. In this case, I repeat $2$ in each suit.



Is the above correct? Or I miss some key assumptions?










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Yes, in general the permutation is idempotent when is a disjoint product of fix points and cycles of length $2.$
    $endgroup$
    – Phicar
    4 hours ago






  • 1




    $begingroup$
    Yes, it's correct. A permutation matrix describes a permutation $pi$. You want $E^2 = I$, so $picircpi = id$.
    $endgroup$
    – amsmath
    4 hours ago
















1












$begingroup$


I am now playing with permutation matrices, http://mathworld.wolfram.com/PermutationMatrix.html.



Also, there is a similar discussion: Symmetric Permutation Matrix.
I want to ask more details than this one.



As we know, a permutation matrix is orthogonal, i.e., $E^T=E^{-1}$. I am interested in when it is symmetric, i.e., $E^T=E^{-1} = E$



Suppose




  1. Start from an identity matrix $I_n$.


  2. $n$ can be even or odd number.

  3. Pick $(i,j)$, where $0<i,jleq n$ and $i, j$ are integer. Exchange $i$-th and $j$-th columns of $I_n$ (identity matrix) and get $E$. Then $E$ is symmetric. This is because $E_{ii}=E_{jj}=0$ and $E_{ij}=E_{ji}=1$.

  4. Based on 3., if I pick a number of sets $(i,j)$, $(k,l)$, $(q,r), ldots$, without repeated index in each $(cdot,cdot)$, and permute columns of $I_n$ according to these sets, then the resulting permutation matrix $E$ is symmetric.


One key thing here is "without repeated index in each $(cdot,cdot)$". This is because if I do $(1,2)$ and $(2,3)$ for $I_3$ for example, I get



$$begin{bmatrix}0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 end{bmatrix},$$



which is not symmetric. In this case, I repeat $2$ in each suit.



Is the above correct? Or I miss some key assumptions?










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Yes, in general the permutation is idempotent when is a disjoint product of fix points and cycles of length $2.$
    $endgroup$
    – Phicar
    4 hours ago






  • 1




    $begingroup$
    Yes, it's correct. A permutation matrix describes a permutation $pi$. You want $E^2 = I$, so $picircpi = id$.
    $endgroup$
    – amsmath
    4 hours ago














1












1








1





$begingroup$


I am now playing with permutation matrices, http://mathworld.wolfram.com/PermutationMatrix.html.



Also, there is a similar discussion: Symmetric Permutation Matrix.
I want to ask more details than this one.



As we know, a permutation matrix is orthogonal, i.e., $E^T=E^{-1}$. I am interested in when it is symmetric, i.e., $E^T=E^{-1} = E$



Suppose




  1. Start from an identity matrix $I_n$.


  2. $n$ can be even or odd number.

  3. Pick $(i,j)$, where $0<i,jleq n$ and $i, j$ are integer. Exchange $i$-th and $j$-th columns of $I_n$ (identity matrix) and get $E$. Then $E$ is symmetric. This is because $E_{ii}=E_{jj}=0$ and $E_{ij}=E_{ji}=1$.

  4. Based on 3., if I pick a number of sets $(i,j)$, $(k,l)$, $(q,r), ldots$, without repeated index in each $(cdot,cdot)$, and permute columns of $I_n$ according to these sets, then the resulting permutation matrix $E$ is symmetric.


One key thing here is "without repeated index in each $(cdot,cdot)$". This is because if I do $(1,2)$ and $(2,3)$ for $I_3$ for example, I get



$$begin{bmatrix}0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 end{bmatrix},$$



which is not symmetric. In this case, I repeat $2$ in each suit.



Is the above correct? Or I miss some key assumptions?










share|cite|improve this question









$endgroup$




I am now playing with permutation matrices, http://mathworld.wolfram.com/PermutationMatrix.html.



Also, there is a similar discussion: Symmetric Permutation Matrix.
I want to ask more details than this one.



As we know, a permutation matrix is orthogonal, i.e., $E^T=E^{-1}$. I am interested in when it is symmetric, i.e., $E^T=E^{-1} = E$



Suppose




  1. Start from an identity matrix $I_n$.


  2. $n$ can be even or odd number.

  3. Pick $(i,j)$, where $0<i,jleq n$ and $i, j$ are integer. Exchange $i$-th and $j$-th columns of $I_n$ (identity matrix) and get $E$. Then $E$ is symmetric. This is because $E_{ii}=E_{jj}=0$ and $E_{ij}=E_{ji}=1$.

  4. Based on 3., if I pick a number of sets $(i,j)$, $(k,l)$, $(q,r), ldots$, without repeated index in each $(cdot,cdot)$, and permute columns of $I_n$ according to these sets, then the resulting permutation matrix $E$ is symmetric.


One key thing here is "without repeated index in each $(cdot,cdot)$". This is because if I do $(1,2)$ and $(2,3)$ for $I_3$ for example, I get



$$begin{bmatrix}0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 end{bmatrix},$$



which is not symmetric. In this case, I repeat $2$ in each suit.



Is the above correct? Or I miss some key assumptions?







linear-algebra matrices permutations






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asked 4 hours ago









sleeve chensleeve chen

3,20042256




3,20042256








  • 1




    $begingroup$
    Yes, in general the permutation is idempotent when is a disjoint product of fix points and cycles of length $2.$
    $endgroup$
    – Phicar
    4 hours ago






  • 1




    $begingroup$
    Yes, it's correct. A permutation matrix describes a permutation $pi$. You want $E^2 = I$, so $picircpi = id$.
    $endgroup$
    – amsmath
    4 hours ago














  • 1




    $begingroup$
    Yes, in general the permutation is idempotent when is a disjoint product of fix points and cycles of length $2.$
    $endgroup$
    – Phicar
    4 hours ago






  • 1




    $begingroup$
    Yes, it's correct. A permutation matrix describes a permutation $pi$. You want $E^2 = I$, so $picircpi = id$.
    $endgroup$
    – amsmath
    4 hours ago








1




1




$begingroup$
Yes, in general the permutation is idempotent when is a disjoint product of fix points and cycles of length $2.$
$endgroup$
– Phicar
4 hours ago




$begingroup$
Yes, in general the permutation is idempotent when is a disjoint product of fix points and cycles of length $2.$
$endgroup$
– Phicar
4 hours ago




1




1




$begingroup$
Yes, it's correct. A permutation matrix describes a permutation $pi$. You want $E^2 = I$, so $picircpi = id$.
$endgroup$
– amsmath
4 hours ago




$begingroup$
Yes, it's correct. A permutation matrix describes a permutation $pi$. You want $E^2 = I$, so $picircpi = id$.
$endgroup$
– amsmath
4 hours ago










2 Answers
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$begingroup$

You’re correct!



We can think of the action of $E$ on the set of $n$ standard basis vectors as a permutation $sigma$ on ${1,dots,n}$ and vice versa.



Let $E$ be symmetric, and let $i$ be the only nonzero entry in the first row. This means that $e_{1i}=e_{i1}$ by symmetry. Thus $E$ swaps the first and the $i^{th}$ standard basis vectors, so $(1~i)$ is a cycle in the cycle decomposition of $sigma$. This argument applies to the rest of the rows to show that $sigma$ is a product of disjoint transpositions.






share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    As you have noted condition for a permutation matrix $E$ to be symmetric
    is that $E^{-1}=E$, and this condition can be expressed as $E^2=I$.



    Interpreting the last condition as repeating the permutation is identity. So this represents a permutation that is its own inverse. That is if $E$ sends a basis vector $v$ to $W$ $E^2=I$ implies $Ew=v$. (possible that $v=w$)



    So this corresponds to a permutation where an element is fixed, or if it sends $x$ to $y$ then it has to send $y$ to $x$. Thus this consists of many disjoint swaps (and possibly some fixed points).



    In group theory it is a permutation of cycle type corresponding to the partition of $n$ into $2$'s and $1$'s. For example $9=2+2+2+ 1^6 $ (that is 1 repeated six times).






    share|cite|improve this answer









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      2 Answers
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      2 Answers
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      $begingroup$

      You’re correct!



      We can think of the action of $E$ on the set of $n$ standard basis vectors as a permutation $sigma$ on ${1,dots,n}$ and vice versa.



      Let $E$ be symmetric, and let $i$ be the only nonzero entry in the first row. This means that $e_{1i}=e_{i1}$ by symmetry. Thus $E$ swaps the first and the $i^{th}$ standard basis vectors, so $(1~i)$ is a cycle in the cycle decomposition of $sigma$. This argument applies to the rest of the rows to show that $sigma$ is a product of disjoint transpositions.






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        You’re correct!



        We can think of the action of $E$ on the set of $n$ standard basis vectors as a permutation $sigma$ on ${1,dots,n}$ and vice versa.



        Let $E$ be symmetric, and let $i$ be the only nonzero entry in the first row. This means that $e_{1i}=e_{i1}$ by symmetry. Thus $E$ swaps the first and the $i^{th}$ standard basis vectors, so $(1~i)$ is a cycle in the cycle decomposition of $sigma$. This argument applies to the rest of the rows to show that $sigma$ is a product of disjoint transpositions.






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          You’re correct!



          We can think of the action of $E$ on the set of $n$ standard basis vectors as a permutation $sigma$ on ${1,dots,n}$ and vice versa.



          Let $E$ be symmetric, and let $i$ be the only nonzero entry in the first row. This means that $e_{1i}=e_{i1}$ by symmetry. Thus $E$ swaps the first and the $i^{th}$ standard basis vectors, so $(1~i)$ is a cycle in the cycle decomposition of $sigma$. This argument applies to the rest of the rows to show that $sigma$ is a product of disjoint transpositions.






          share|cite|improve this answer









          $endgroup$



          You’re correct!



          We can think of the action of $E$ on the set of $n$ standard basis vectors as a permutation $sigma$ on ${1,dots,n}$ and vice versa.



          Let $E$ be symmetric, and let $i$ be the only nonzero entry in the first row. This means that $e_{1i}=e_{i1}$ by symmetry. Thus $E$ swaps the first and the $i^{th}$ standard basis vectors, so $(1~i)$ is a cycle in the cycle decomposition of $sigma$. This argument applies to the rest of the rows to show that $sigma$ is a product of disjoint transpositions.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 2 hours ago









          Santana AftonSantana Afton

          3,1922730




          3,1922730























              2












              $begingroup$

              As you have noted condition for a permutation matrix $E$ to be symmetric
              is that $E^{-1}=E$, and this condition can be expressed as $E^2=I$.



              Interpreting the last condition as repeating the permutation is identity. So this represents a permutation that is its own inverse. That is if $E$ sends a basis vector $v$ to $W$ $E^2=I$ implies $Ew=v$. (possible that $v=w$)



              So this corresponds to a permutation where an element is fixed, or if it sends $x$ to $y$ then it has to send $y$ to $x$. Thus this consists of many disjoint swaps (and possibly some fixed points).



              In group theory it is a permutation of cycle type corresponding to the partition of $n$ into $2$'s and $1$'s. For example $9=2+2+2+ 1^6 $ (that is 1 repeated six times).






              share|cite|improve this answer









              $endgroup$


















                2












                $begingroup$

                As you have noted condition for a permutation matrix $E$ to be symmetric
                is that $E^{-1}=E$, and this condition can be expressed as $E^2=I$.



                Interpreting the last condition as repeating the permutation is identity. So this represents a permutation that is its own inverse. That is if $E$ sends a basis vector $v$ to $W$ $E^2=I$ implies $Ew=v$. (possible that $v=w$)



                So this corresponds to a permutation where an element is fixed, or if it sends $x$ to $y$ then it has to send $y$ to $x$. Thus this consists of many disjoint swaps (and possibly some fixed points).



                In group theory it is a permutation of cycle type corresponding to the partition of $n$ into $2$'s and $1$'s. For example $9=2+2+2+ 1^6 $ (that is 1 repeated six times).






                share|cite|improve this answer









                $endgroup$
















                  2












                  2








                  2





                  $begingroup$

                  As you have noted condition for a permutation matrix $E$ to be symmetric
                  is that $E^{-1}=E$, and this condition can be expressed as $E^2=I$.



                  Interpreting the last condition as repeating the permutation is identity. So this represents a permutation that is its own inverse. That is if $E$ sends a basis vector $v$ to $W$ $E^2=I$ implies $Ew=v$. (possible that $v=w$)



                  So this corresponds to a permutation where an element is fixed, or if it sends $x$ to $y$ then it has to send $y$ to $x$. Thus this consists of many disjoint swaps (and possibly some fixed points).



                  In group theory it is a permutation of cycle type corresponding to the partition of $n$ into $2$'s and $1$'s. For example $9=2+2+2+ 1^6 $ (that is 1 repeated six times).






                  share|cite|improve this answer









                  $endgroup$



                  As you have noted condition for a permutation matrix $E$ to be symmetric
                  is that $E^{-1}=E$, and this condition can be expressed as $E^2=I$.



                  Interpreting the last condition as repeating the permutation is identity. So this represents a permutation that is its own inverse. That is if $E$ sends a basis vector $v$ to $W$ $E^2=I$ implies $Ew=v$. (possible that $v=w$)



                  So this corresponds to a permutation where an element is fixed, or if it sends $x$ to $y$ then it has to send $y$ to $x$. Thus this consists of many disjoint swaps (and possibly some fixed points).



                  In group theory it is a permutation of cycle type corresponding to the partition of $n$ into $2$'s and $1$'s. For example $9=2+2+2+ 1^6 $ (that is 1 repeated six times).







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 2 hours ago









                  P VanchinathanP Vanchinathan

                  15.7k12236




                  15.7k12236






























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