Eigenvalues and Eigenvectors of diagonal marix
$begingroup$
Problem:
Let $D$:= diag($lambda_1, ldots, lambda_n$), i.e., $D$ is a diagonal matrix in $mathbb{C}^{ntimes n}$ with entries $lambda_1, ldots, lambda_n$ ∈ $mathbb{C}$ on its diagonal.
Find $sigma$($D$) and all eigenvectors of $D$.
My thoughts:
As the spectrum of $D$ is the set of all eigenvalues, then $sigma$($D$) should be just $lambda_1 cdots lambda_n$ = $mathbb {lambda_n}^{n}$ .
But how can I find the eigenvectors ? I know I have to calculate the $D$ - $lambda I$.
Can someone help me?
linear-algebra eigenvalues-eigenvectors
edited Jan 6 at 9:31
Berkheimer
1,4371024
asked Jan 6 at 9:21
KaiKai
636
$endgroup$
add a comment |
$begingroup$
Problem:
Let $D$:= diag($lambda_1, ldots, lambda_n$), i.e., $D$ is a diagonal matrix in $mathbb{C}^{ntimes n}$ with entries $lambda_1, ldots, lambda_n$ ∈ $mathbb{C}$ on its diagonal.
Find $sigma$($D$) and all eigenvectors of $D$.
My thoughts:
As the spectrum of $D$ is the set of all eigenvalues, then $sigma$($D$) should be just $lambda_1 cdots lambda_n$ = $mathbb {lambda_n}^{n}$ .
But how can I find the eigenvectors ? I know I have to calculate the $D$ - $lambda I$.
Can someone help me?
linear-algebra eigenvalues-eigenvectors
edited Jan 6 at 9:31
Berkheimer
1,4371024
asked Jan 6 at 9:21
KaiKai
636
$endgroup$
add a comment |
$begingroup$
Problem:
Let $D$:= diag($lambda_1, ldots, lambda_n$), i.e., $D$ is a diagonal matrix in $mathbb{C}^{ntimes n}$ with entries $lambda_1, ldots, lambda_n$ ∈ $mathbb{C}$ on its diagonal.
Find $sigma$($D$) and all eigenvectors of $D$.
My thoughts:
As the spectrum of $D$ is the set of all eigenvalues, then $sigma$($D$) should be just $lambda_1 cdots lambda_n$ = $mathbb {lambda_n}^{n}$ .
But how can I find the eigenvectors ? I know I have to calculate the $D$ - $lambda I$.
Can someone help me?
linear-algebra eigenvalues-eigenvectors
edited Jan 6 at 9:31
Berkheimer
1,4371024
asked Jan 6 at 9:21
KaiKai
636
$endgroup$
Problem:
Let $D$:= diag($lambda_1, ldots, lambda_n$), i.e., $D$ is a diagonal matrix in $mathbb{C}^{ntimes n}$ with entries $lambda_1, ldots, lambda_n$ ∈ $mathbb{C}$ on its diagonal.
Find $sigma$($D$) and all eigenvectors of $D$.
My thoughts:
As the spectrum of $D$ is the set of all eigenvalues, then $sigma$($D$) should be just $lambda_1 cdots lambda_n$ = $mathbb {lambda_n}^{n}$ .
But how can I find the eigenvectors ? I know I have to calculate the $D$ - $lambda I$.
Can someone help me?
linear-algebra eigenvalues-eigenvectors
linear-algebra eigenvalues-eigenvectors
edited Jan 6 at 9:31
Berkheimer
1,4371024
asked Jan 6 at 9:21
KaiKai
636
edited Jan 6 at 9:31
Berkheimer
1,4371024
asked Jan 6 at 9:21
KaiKai
636
edited Jan 6 at 9:31
Berkheimer
1,4371024
edited Jan 6 at 9:31
Berkheimer
1,4371024
edited Jan 6 at 9:31
Berkheimer
1,4371024
1,4371024
asked Jan 6 at 9:21
KaiKai
636
asked Jan 6 at 9:21
KaiKai
636
asked Jan 6 at 9:21
KaiKai
636
636
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If the matrix is diagonal, the eigenvectors are just the standard basis of $mathbb{C}$:
$$
e_1 = (1, 0, dots, 0)^t, dots, e_n=(0,dots, 0, 1)^t .
$$
answered Jan 6 at 9:33
d.t.d.t.
14.2k23074
$endgroup$
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%2f3063639%2feigenvalues-and-eigenvectors-of-diagonal-marix%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If the matrix is diagonal, the eigenvectors are just the standard basis of $mathbb{C}$:
$$
e_1 = (1, 0, dots, 0)^t, dots, e_n=(0,dots, 0, 1)^t .
$$
answered Jan 6 at 9:33
d.t.d.t.
14.2k23074
$endgroup$
add a comment |
$begingroup$
If the matrix is diagonal, the eigenvectors are just the standard basis of $mathbb{C}$:
$$
e_1 = (1, 0, dots, 0)^t, dots, e_n=(0,dots, 0, 1)^t .
$$
answered Jan 6 at 9:33
d.t.d.t.
14.2k23074
$endgroup$
add a comment |
$begingroup$
If the matrix is diagonal, the eigenvectors are just the standard basis of $mathbb{C}$:
$$
e_1 = (1, 0, dots, 0)^t, dots, e_n=(0,dots, 0, 1)^t .
$$
answered Jan 6 at 9:33
d.t.d.t.
14.2k23074
$endgroup$
If the matrix is diagonal, the eigenvectors are just the standard basis of $mathbb{C}$:
$$
e_1 = (1, 0, dots, 0)^t, dots, e_n=(0,dots, 0, 1)^t .
$$
answered Jan 6 at 9:33
d.t.d.t.
14.2k23074
answered Jan 6 at 9:33
d.t.d.t.
14.2k23074
answered Jan 6 at 9:33
d.t.d.t.
14.2k23074
answered Jan 6 at 9:33
d.t.d.t.
14.2k23074
14.2k23074
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%2f3063639%2feigenvalues-and-eigenvectors-of-diagonal-marix%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
