Proof that $M$ and $M^{T}$ are similar












1












$begingroup$



Triangular matrix $M in mathbb R^{n,n}$ and all elements on the diagonal are different. Proof that $M$ and $M^{T}$ are similar.





I know that the matrices are similar when the matrix similarity relation is the relation of equivalence so it is reflexive, symmetric and transitive relation.Unfortunately I don't know what to use this information in my task.


Can you get some tips?






linear-algebra







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  • 3




    $begingroup$
    $M$ is diagonalizable, that is $P^{-1}MP = D$
    $endgroup$
    – Will Jagy
    Feb 2 at 22:40
















1












$begingroup$



Triangular matrix $M in mathbb R^{n,n}$ and all elements on the diagonal are different. Proof that $M$ and $M^{T}$ are similar.





I know that the matrices are similar when the matrix similarity relation is the relation of equivalence so it is reflexive, symmetric and transitive relation.Unfortunately I don't know what to use this information in my task.


Can you get some tips?






linear-algebra







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    $M$ is diagonalizable, that is $P^{-1}MP = D$
    $endgroup$
    – Will Jagy
    Feb 2 at 22:40














1












1








1





$begingroup$



Triangular matrix $M in mathbb R^{n,n}$ and all elements on the diagonal are different. Proof that $M$ and $M^{T}$ are similar.





I know that the matrices are similar when the matrix similarity relation is the relation of equivalence so it is reflexive, symmetric and transitive relation.Unfortunately I don't know what to use this information in my task.


Can you get some tips?






linear-algebra







share|cite|improve this question









$endgroup$





Triangular matrix $M in mathbb R^{n,n}$ and all elements on the diagonal are different. Proof that $M$ and $M^{T}$ are similar.





I know that the matrices are similar when the matrix similarity relation is the relation of equivalence so it is reflexive, symmetric and transitive relation.Unfortunately I don't know what to use this information in my task.


Can you get some tips?






linear-algebra




linear-algebra






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share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Feb 2 at 22:35









VirtualUserVirtualUser

1,321317




1,321317








  • 3




    $begingroup$
    $M$ is diagonalizable, that is $P^{-1}MP = D$
    $endgroup$
    – Will Jagy
    Feb 2 at 22:40














  • 3




    $begingroup$
    $M$ is diagonalizable, that is $P^{-1}MP = D$
    $endgroup$
    – Will Jagy
    Feb 2 at 22:40








3




3




$begingroup$
$M$ is diagonalizable, that is $P^{-1}MP = D$
$endgroup$
– Will Jagy
Feb 2 at 22:40




$begingroup$
$M$ is diagonalizable, that is $P^{-1}MP = D$
$endgroup$
– Will Jagy
Feb 2 at 22:40










1 Answer
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$begingroup$

Presume that they are similar, and in particular $M = P^{-1}M^{T}P$. What can you say about $P$?



Then write the equation as $PM = M^{T}P$. Given what we know about $P$, this should suggest what we need for $M$ and $M^{T}$ to be similar (i.e. it should tell us what type of matrix $P$ is sufficient to make the equation true).






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    $begingroup$

    Presume that they are similar, and in particular $M = P^{-1}M^{T}P$. What can you say about $P$?



    Then write the equation as $PM = M^{T}P$. Given what we know about $P$, this should suggest what we need for $M$ and $M^{T}$ to be similar (i.e. it should tell us what type of matrix $P$ is sufficient to make the equation true).






    share|cite|improve this answer









    $endgroup$


















      3












      $begingroup$

      Presume that they are similar, and in particular $M = P^{-1}M^{T}P$. What can you say about $P$?



      Then write the equation as $PM = M^{T}P$. Given what we know about $P$, this should suggest what we need for $M$ and $M^{T}$ to be similar (i.e. it should tell us what type of matrix $P$ is sufficient to make the equation true).






      share|cite|improve this answer









      $endgroup$
















        3












        3








        3





        $begingroup$

        Presume that they are similar, and in particular $M = P^{-1}M^{T}P$. What can you say about $P$?



        Then write the equation as $PM = M^{T}P$. Given what we know about $P$, this should suggest what we need for $M$ and $M^{T}$ to be similar (i.e. it should tell us what type of matrix $P$ is sufficient to make the equation true).






        share|cite|improve this answer









        $endgroup$



        Presume that they are similar, and in particular $M = P^{-1}M^{T}P$. What can you say about $P$?



        Then write the equation as $PM = M^{T}P$. Given what we know about $P$, this should suggest what we need for $M$ and $M^{T}$ to be similar (i.e. it should tell us what type of matrix $P$ is sufficient to make the equation true).







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Feb 2 at 22:44









        Jacob MaibachJacob Maibach

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