Does exists a matrix $X$ for how many $n$ such that $X^n=A$?
$begingroup$
Let
$$
A= begin{pmatrix}
0 & 1 & 2 \
0 & 0 & 1 \
0 & 0 & 0 \
end{pmatrix}
$$
For how many $n$ is there a matrix $X$ such that $X^n=A$?
matrix-calculus
edited Dec 19 '18 at 18:51
user1101010
9011830
asked Dec 19 '18 at 18:25
Lucas DantasLucas Dantas
41
$endgroup$
add a comment |
$begingroup$
Let
$$
A= begin{pmatrix}
0 & 1 & 2 \
0 & 0 & 1 \
0 & 0 & 0 \
end{pmatrix}
$$
For how many $n$ is there a matrix $X$ such that $X^n=A$?
matrix-calculus
edited Dec 19 '18 at 18:51
user1101010
9011830
asked Dec 19 '18 at 18:25
Lucas DantasLucas Dantas
41
$endgroup$
$begingroup$
Do you know what nilpotent matrices / endomorphisms are?
$endgroup$
– Viktor Glombik
Dec 19 '18 at 18:31
add a comment |
$begingroup$
Let
$$
A= begin{pmatrix}
0 & 1 & 2 \
0 & 0 & 1 \
0 & 0 & 0 \
end{pmatrix}
$$
For how many $n$ is there a matrix $X$ such that $X^n=A$?
matrix-calculus
edited Dec 19 '18 at 18:51
user1101010
9011830
asked Dec 19 '18 at 18:25
Lucas DantasLucas Dantas
41
$endgroup$
Let
$$
A= begin{pmatrix}
0 & 1 & 2 \
0 & 0 & 1 \
0 & 0 & 0 \
end{pmatrix}
$$
For how many $n$ is there a matrix $X$ such that $X^n=A$?
matrix-calculus
matrix-calculus
edited Dec 19 '18 at 18:51
user1101010
9011830
asked Dec 19 '18 at 18:25
Lucas DantasLucas Dantas
41
edited Dec 19 '18 at 18:51
user1101010
9011830
asked Dec 19 '18 at 18:25
Lucas DantasLucas Dantas
41
edited Dec 19 '18 at 18:51
user1101010
9011830
edited Dec 19 '18 at 18:51
user1101010
9011830
edited Dec 19 '18 at 18:51
user1101010
9011830
9011830
asked Dec 19 '18 at 18:25
Lucas DantasLucas Dantas
41
asked Dec 19 '18 at 18:25
Lucas DantasLucas Dantas
41
asked Dec 19 '18 at 18:25
Lucas DantasLucas Dantas
41
41
$begingroup$
Do you know what nilpotent matrices / endomorphisms are?
$endgroup$
– Viktor Glombik
Dec 19 '18 at 18:31
add a comment |
$begingroup$
Do you know what nilpotent matrices / endomorphisms are?
$endgroup$
– Viktor Glombik
Dec 19 '18 at 18:31
$begingroup$
Do you know what nilpotent matrices / endomorphisms are?
$endgroup$
– Viktor Glombik
Dec 19 '18 at 18:31
$begingroup$
Do you know what nilpotent matrices / endomorphisms are?
$endgroup$
– Viktor Glombik
Dec 19 '18 at 18:31
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Hint: what could be the minimal polynomial of such an $X$?
answered Dec 19 '18 at 18:32
Robert IsraelRobert Israel
330k23219473
$endgroup$
add a comment |
$begingroup$
For how many $n$? Only for finitely many $n$. More precisely only for $nle 2$, since $X$ must be nilpotent because of $A^2=0$ and $X^n=A$. However, a nilpotent matrix $Xin M_3(K)$ satisfies $X^3=0$. It follows that $X^n=0neq A$ for all $nge 3$.
answered Dec 19 '18 at 19:13
Dietrich BurdeDietrich Burde
81.6k648106
$endgroup$
add a comment |
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%2f3046704%2fdoes-exists-a-matrix-x-for-how-many-n-such-that-xn-a%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: what could be the minimal polynomial of such an $X$?
answered Dec 19 '18 at 18:32
Robert IsraelRobert Israel
330k23219473
$endgroup$
add a comment |
$begingroup$
Hint: what could be the minimal polynomial of such an $X$?
answered Dec 19 '18 at 18:32
Robert IsraelRobert Israel
330k23219473
$endgroup$
add a comment |
$begingroup$
Hint: what could be the minimal polynomial of such an $X$?
answered Dec 19 '18 at 18:32
Robert IsraelRobert Israel
330k23219473
$endgroup$
Hint: what could be the minimal polynomial of such an $X$?
answered Dec 19 '18 at 18:32
Robert IsraelRobert Israel
330k23219473
answered Dec 19 '18 at 18:32
Robert IsraelRobert Israel
330k23219473
answered Dec 19 '18 at 18:32
Robert IsraelRobert Israel
330k23219473
answered Dec 19 '18 at 18:32
Robert IsraelRobert Israel
330k23219473
330k23219473
add a comment |
add a comment |
$begingroup$
For how many $n$? Only for finitely many $n$. More precisely only for $nle 2$, since $X$ must be nilpotent because of $A^2=0$ and $X^n=A$. However, a nilpotent matrix $Xin M_3(K)$ satisfies $X^3=0$. It follows that $X^n=0neq A$ for all $nge 3$.
answered Dec 19 '18 at 19:13
Dietrich BurdeDietrich Burde
81.6k648106
$endgroup$
add a comment |
$begingroup$
For how many $n$? Only for finitely many $n$. More precisely only for $nle 2$, since $X$ must be nilpotent because of $A^2=0$ and $X^n=A$. However, a nilpotent matrix $Xin M_3(K)$ satisfies $X^3=0$. It follows that $X^n=0neq A$ for all $nge 3$.
answered Dec 19 '18 at 19:13
Dietrich BurdeDietrich Burde
81.6k648106
$endgroup$
add a comment |
$begingroup$
For how many $n$? Only for finitely many $n$. More precisely only for $nle 2$, since $X$ must be nilpotent because of $A^2=0$ and $X^n=A$. However, a nilpotent matrix $Xin M_3(K)$ satisfies $X^3=0$. It follows that $X^n=0neq A$ for all $nge 3$.
answered Dec 19 '18 at 19:13
Dietrich BurdeDietrich Burde
81.6k648106
$endgroup$
For how many $n$? Only for finitely many $n$. More precisely only for $nle 2$, since $X$ must be nilpotent because of $A^2=0$ and $X^n=A$. However, a nilpotent matrix $Xin M_3(K)$ satisfies $X^3=0$. It follows that $X^n=0neq A$ for all $nge 3$.
answered Dec 19 '18 at 19:13
Dietrich BurdeDietrich Burde
81.6k648106
answered Dec 19 '18 at 19:13
Dietrich BurdeDietrich Burde
81.6k648106
answered Dec 19 '18 at 19:13
Dietrich BurdeDietrich Burde
81.6k648106
answered Dec 19 '18 at 19:13
Dietrich BurdeDietrich Burde
81.6k648106
81.6k648106
add a comment |
add a comment |
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%2f3046704%2fdoes-exists-a-matrix-x-for-how-many-n-such-that-xn-a%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
$begingroup$
Do you know what nilpotent matrices / endomorphisms are?
$endgroup$
– Viktor Glombik
Dec 19 '18 at 18:31