Nilpotent Operators Linear Algebra
Let $V$ be an $n$-dimensional vector space over an arbitrary field $K$, and let
$T_1, dots , T_n : V rightarrow V$ be pairwise commuting nilpotent operators on V
(a) Show the composition $T_1 T_2 · · · T_n = 0$.
(b) Does this still hold if we drop the hypothesis that the $T_i$ are pairwise commuting?
What I have tried:
I know that for any $T_i$, $ker({T_i}^n) = ker({T_i}^{n+1}) = cdots$. This implies that ${T_i}^n = 0$ for any $i$. I also know that raising each $T_i$ to a power of itself lessens the dimension of the kernel of such a composition. Any help?
nilpotence
add a comment |
Let $V$ be an $n$-dimensional vector space over an arbitrary field $K$, and let
$T_1, dots , T_n : V rightarrow V$ be pairwise commuting nilpotent operators on V
(a) Show the composition $T_1 T_2 · · · T_n = 0$.
(b) Does this still hold if we drop the hypothesis that the $T_i$ are pairwise commuting?
What I have tried:
I know that for any $T_i$, $ker({T_i}^n) = ker({T_i}^{n+1}) = cdots$. This implies that ${T_i}^n = 0$ for any $i$. I also know that raising each $T_i$ to a power of itself lessens the dimension of the kernel of such a composition. Any help?
nilpotence
add a comment |
Let $V$ be an $n$-dimensional vector space over an arbitrary field $K$, and let
$T_1, dots , T_n : V rightarrow V$ be pairwise commuting nilpotent operators on V
(a) Show the composition $T_1 T_2 · · · T_n = 0$.
(b) Does this still hold if we drop the hypothesis that the $T_i$ are pairwise commuting?
What I have tried:
I know that for any $T_i$, $ker({T_i}^n) = ker({T_i}^{n+1}) = cdots$. This implies that ${T_i}^n = 0$ for any $i$. I also know that raising each $T_i$ to a power of itself lessens the dimension of the kernel of such a composition. Any help?
nilpotence
Let $V$ be an $n$-dimensional vector space over an arbitrary field $K$, and let
$T_1, dots , T_n : V rightarrow V$ be pairwise commuting nilpotent operators on V
(a) Show the composition $T_1 T_2 · · · T_n = 0$.
(b) Does this still hold if we drop the hypothesis that the $T_i$ are pairwise commuting?
What I have tried:
I know that for any $T_i$, $ker({T_i}^n) = ker({T_i}^{n+1}) = cdots$. This implies that ${T_i}^n = 0$ for any $i$. I also know that raising each $T_i$ to a power of itself lessens the dimension of the kernel of such a composition. Any help?
nilpotence
nilpotence
edited Dec 10 '18 at 20:50
asked Dec 9 '18 at 23:21
user624358
add a comment |
add a comment |
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%2f3033175%2fnilpotent-operators-linear-algebra%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3033175%2fnilpotent-operators-linear-algebra%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
