Eigenspaces of an orthogonal projection
up vote
1
down vote
favorite
So for an orthogonal projection $P:Vrightarrow U $, the task is to find the eigenvalues and eigenspaces of P.
I have found that $lambda = 0,1$
Then $E_0 = ker(P) = U^{perp}$
but for $E_1$ i'm not sure
$E_1 = ker(P-Id)$
I feel like the solution for this is somewhat trivial and is staring me in the face, but i cant see it
Can anyone explain? thanks
eigenvalues-eigenvectors
asked Nov 22 at 4:20
big daddy
434
add a comment |
up vote
1
down vote
favorite
So for an orthogonal projection $P:Vrightarrow U $, the task is to find the eigenvalues and eigenspaces of P.
I have found that $lambda = 0,1$
Then $E_0 = ker(P) = U^{perp}$
but for $E_1$ i'm not sure
$E_1 = ker(P-Id)$
I feel like the solution for this is somewhat trivial and is staring me in the face, but i cant see it
Can anyone explain? thanks
eigenvalues-eigenvectors
asked Nov 22 at 4:20
big daddy
434
1
What actually do you want to find?
– Anupam
Nov 22 at 4:30
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
So for an orthogonal projection $P:Vrightarrow U $, the task is to find the eigenvalues and eigenspaces of P.
I have found that $lambda = 0,1$
Then $E_0 = ker(P) = U^{perp}$
but for $E_1$ i'm not sure
$E_1 = ker(P-Id)$
I feel like the solution for this is somewhat trivial and is staring me in the face, but i cant see it
Can anyone explain? thanks
eigenvalues-eigenvectors
asked Nov 22 at 4:20
big daddy
434
So for an orthogonal projection $P:Vrightarrow U $, the task is to find the eigenvalues and eigenspaces of P.
I have found that $lambda = 0,1$
Then $E_0 = ker(P) = U^{perp}$
but for $E_1$ i'm not sure
$E_1 = ker(P-Id)$
I feel like the solution for this is somewhat trivial and is staring me in the face, but i cant see it
Can anyone explain? thanks
eigenvalues-eigenvectors
eigenvalues-eigenvectors
asked Nov 22 at 4:20
big daddy
434
asked Nov 22 at 4:20
big daddy
434
asked Nov 22 at 4:20
big daddy
434
asked Nov 22 at 4:20
big daddy
434
asked Nov 22 at 4:20
big daddy
434
434
1
What actually do you want to find?
– Anupam
Nov 22 at 4:30
add a comment |
1
What actually do you want to find?
– Anupam
Nov 22 at 4:30
1
1
What actually do you want to find?
– Anupam
Nov 22 at 4:30
What actually do you want to find?
– Anupam
Nov 22 at 4:30
add a comment |
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
Let $x in E_1$. Then, $x=P(x) in P(V)$. Hence, $E_1 subseteq P(V)$.
Let, $y in P(V)$. Then, $y=P(X)$ for some $x in V$. Hence, $P(y)=P^2(x)=P(x)=y$ i.e. $y in E_1$. $therefore P(V) subseteq E_1$.
Combining, we get, $P(V)=E_1$.
answered Nov 22 at 5:51
SinTan1729
2,399622
add a comment |
up vote
1
down vote
$E_1$ can be proved to be the range of P. You'll have to do a bit of work, but it's not too hard to show.
answered Nov 22 at 5:05
JonathanZ
2,089613
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',
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%2f3008736%2feigenspaces-of-an-orthogonal-projection%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
up vote
1
down vote
accepted
Let $x in E_1$. Then, $x=P(x) in P(V)$. Hence, $E_1 subseteq P(V)$.
Let, $y in P(V)$. Then, $y=P(X)$ for some $x in V$. Hence, $P(y)=P^2(x)=P(x)=y$ i.e. $y in E_1$. $therefore P(V) subseteq E_1$.
Combining, we get, $P(V)=E_1$.
answered Nov 22 at 5:51
SinTan1729
2,399622
add a comment |
up vote
1
down vote
accepted
Let $x in E_1$. Then, $x=P(x) in P(V)$. Hence, $E_1 subseteq P(V)$.
Let, $y in P(V)$. Then, $y=P(X)$ for some $x in V$. Hence, $P(y)=P^2(x)=P(x)=y$ i.e. $y in E_1$. $therefore P(V) subseteq E_1$.
Combining, we get, $P(V)=E_1$.
answered Nov 22 at 5:51
SinTan1729
2,399622
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Let $x in E_1$. Then, $x=P(x) in P(V)$. Hence, $E_1 subseteq P(V)$.
Let, $y in P(V)$. Then, $y=P(X)$ for some $x in V$. Hence, $P(y)=P^2(x)=P(x)=y$ i.e. $y in E_1$. $therefore P(V) subseteq E_1$.
Combining, we get, $P(V)=E_1$.
answered Nov 22 at 5:51
SinTan1729
2,399622
Let $x in E_1$. Then, $x=P(x) in P(V)$. Hence, $E_1 subseteq P(V)$.
Let, $y in P(V)$. Then, $y=P(X)$ for some $x in V$. Hence, $P(y)=P^2(x)=P(x)=y$ i.e. $y in E_1$. $therefore P(V) subseteq E_1$.
Combining, we get, $P(V)=E_1$.
answered Nov 22 at 5:51
SinTan1729
2,399622
answered Nov 22 at 5:51
SinTan1729
2,399622
answered Nov 22 at 5:51
SinTan1729
2,399622
answered Nov 22 at 5:51
SinTan1729
2,399622
2,399622
add a comment |
add a comment |
up vote
1
down vote
$E_1$ can be proved to be the range of P. You'll have to do a bit of work, but it's not too hard to show.
answered Nov 22 at 5:05
JonathanZ
2,089613
add a comment |
up vote
1
down vote
$E_1$ can be proved to be the range of P. You'll have to do a bit of work, but it's not too hard to show.
answered Nov 22 at 5:05
JonathanZ
2,089613
add a comment |
up vote
1
down vote
up vote
1
down vote
$E_1$ can be proved to be the range of P. You'll have to do a bit of work, but it's not too hard to show.
answered Nov 22 at 5:05
JonathanZ
2,089613
$E_1$ can be proved to be the range of P. You'll have to do a bit of work, but it's not too hard to show.
answered Nov 22 at 5:05
JonathanZ
2,089613
answered Nov 22 at 5:05
JonathanZ
2,089613
answered Nov 22 at 5:05
JonathanZ
2,089613
answered Nov 22 at 5:05
JonathanZ
2,089613
2,089613
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.
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%2f3008736%2feigenspaces-of-an-orthogonal-projection%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
1
What actually do you want to find?
– Anupam
Nov 22 at 4:30