Invariant factors and Jordan reduction : how to find the adapted basis?
$begingroup$
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
asked Jan 3 at 12:40
ChocoSavourChocoSavour
31418
$endgroup$
add a comment |
$begingroup$
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
asked Jan 3 at 12:40
ChocoSavourChocoSavour
31418
$endgroup$
add a comment |
$begingroup$
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
asked Jan 3 at 12:40
ChocoSavourChocoSavour
31418
$endgroup$
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
asked Jan 3 at 12:40
ChocoSavourChocoSavour
31418
asked Jan 3 at 12:40
ChocoSavourChocoSavour
31418
asked Jan 3 at 12:40
ChocoSavourChocoSavour
31418
asked Jan 3 at 12:40
ChocoSavourChocoSavour
31418
asked Jan 3 at 12:40
ChocoSavourChocoSavour
31418
31418
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060526%2finvariant-factors-and-jordan-reduction-how-to-find-the-adapted-basis%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060526%2finvariant-factors-and-jordan-reduction-how-to-find-the-adapted-basis%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
