Invariant factors and Jordan reduction : how to find the adapted basis?












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I have a question about the Jordan reduction using the module theory and especially the invariant factors. If we have a vector space $E$ of dimension $n$ over a field $k$, and $f in End(E)$ then it's also a $k[X]$-module for the operation : $P.v = P(f)(v)$. And of course it's a finitely generated module, and then :



$E cong frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$



as a $k[X]$-module, and $f$ is the multiplication by $X$ for $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$. Then, $E=E_1 bigoplus ... bigoplus E_n$ and on a good basis of E, the matrix of $f$ is the compagnon matrix of $P_i$ on each subspace $E_i$ which is stable by $f$.



($P_i$ can be find as the invariant factors of $det(Mat(f) - XId)$)



And then, if $P_i$ have the form $(X-a)^r$, then we can find a good basis of E when the matrix of $f$ has the Jordan form.



But, this is my problem : how to find the good basis ? Cause on $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$, we know the basis on which the matrix of the multiplication by $X$ have the Jordan form, but actually, to determine it on $E$, we should know, in the explicit way, the isomorphism between $E$ and $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$, and it's seems very complicated, in a practical way...



When we want to apply this principle in a concrete way (for example for the Linear difference equation), we make a change of basis to transform the matrix in order to obtain a matrix which have the jordan form, but we should also know the change of basis if we want to solve the problem, finally...



So, how to do ? And, are we able, from this application of invariant factors, to find the basis adapted to the jordan form ?






linear-algebra abstract-algebra jordan-normal-form







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    0














    I have a question about the Jordan reduction using the module theory and especially the invariant factors. If we have a vector space $E$ of dimension $n$ over a field $k$, and $f in End(E)$ then it's also a $k[X]$-module for the operation : $P.v = P(f)(v)$. And of course it's a finitely generated module, and then :



    $E cong frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$



    as a $k[X]$-module, and $f$ is the multiplication by $X$ for $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$. Then, $E=E_1 bigoplus ... bigoplus E_n$ and on a good basis of E, the matrix of $f$ is the compagnon matrix of $P_i$ on each subspace $E_i$ which is stable by $f$.



    ($P_i$ can be find as the invariant factors of $det(Mat(f) - XId)$)



    And then, if $P_i$ have the form $(X-a)^r$, then we can find a good basis of E when the matrix of $f$ has the Jordan form.



    But, this is my problem : how to find the good basis ? Cause on $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$, we know the basis on which the matrix of the multiplication by $X$ have the Jordan form, but actually, to determine it on $E$, we should know, in the explicit way, the isomorphism between $E$ and $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$, and it's seems very complicated, in a practical way...



    When we want to apply this principle in a concrete way (for example for the Linear difference equation), we make a change of basis to transform the matrix in order to obtain a matrix which have the jordan form, but we should also know the change of basis if we want to solve the problem, finally...



    So, how to do ? And, are we able, from this application of invariant factors, to find the basis adapted to the jordan form ?






    linear-algebra abstract-algebra jordan-normal-form







    share|cite|improve this question

























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      I have a question about the Jordan reduction using the module theory and especially the invariant factors. If we have a vector space $E$ of dimension $n$ over a field $k$, and $f in End(E)$ then it's also a $k[X]$-module for the operation : $P.v = P(f)(v)$. And of course it's a finitely generated module, and then :



      $E cong frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$



      as a $k[X]$-module, and $f$ is the multiplication by $X$ for $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$. Then, $E=E_1 bigoplus ... bigoplus E_n$ and on a good basis of E, the matrix of $f$ is the compagnon matrix of $P_i$ on each subspace $E_i$ which is stable by $f$.



      ($P_i$ can be find as the invariant factors of $det(Mat(f) - XId)$)



      And then, if $P_i$ have the form $(X-a)^r$, then we can find a good basis of E when the matrix of $f$ has the Jordan form.



      But, this is my problem : how to find the good basis ? Cause on $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$, we know the basis on which the matrix of the multiplication by $X$ have the Jordan form, but actually, to determine it on $E$, we should know, in the explicit way, the isomorphism between $E$ and $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$, and it's seems very complicated, in a practical way...



      When we want to apply this principle in a concrete way (for example for the Linear difference equation), we make a change of basis to transform the matrix in order to obtain a matrix which have the jordan form, but we should also know the change of basis if we want to solve the problem, finally...



      So, how to do ? And, are we able, from this application of invariant factors, to find the basis adapted to the jordan form ?






      linear-algebra abstract-algebra jordan-normal-form







      share|cite|improve this question













      I have a question about the Jordan reduction using the module theory and especially the invariant factors. If we have a vector space $E$ of dimension $n$ over a field $k$, and $f in End(E)$ then it's also a $k[X]$-module for the operation : $P.v = P(f)(v)$. And of course it's a finitely generated module, and then :



      $E cong frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$



      as a $k[X]$-module, and $f$ is the multiplication by $X$ for $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$. Then, $E=E_1 bigoplus ... bigoplus E_n$ and on a good basis of E, the matrix of $f$ is the compagnon matrix of $P_i$ on each subspace $E_i$ which is stable by $f$.



      ($P_i$ can be find as the invariant factors of $det(Mat(f) - XId)$)



      And then, if $P_i$ have the form $(X-a)^r$, then we can find a good basis of E when the matrix of $f$ has the Jordan form.



      But, this is my problem : how to find the good basis ? Cause on $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$, we know the basis on which the matrix of the multiplication by $X$ have the Jordan form, but actually, to determine it on $E$, we should know, in the explicit way, the isomorphism between $E$ and $frac{k[X]}{(P_1)} times ... times frac{k[X]}{(P_s)}$, and it's seems very complicated, in a practical way...



      When we want to apply this principle in a concrete way (for example for the Linear difference equation), we make a change of basis to transform the matrix in order to obtain a matrix which have the jordan form, but we should also know the change of basis if we want to solve the problem, finally...



      So, how to do ? And, are we able, from this application of invariant factors, to find the basis adapted to the jordan form ?






      linear-algebra abstract-algebra jordan-normal-form




      linear-algebra abstract-algebra jordan-normal-form






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