Mixing
Do two unitary operators that commute have same eigenvalues?
$begingroup$
I have this example on a textbook but it does not seem to be right.
Example:
If A,B are unitary operators and AB=BA,prove that A and B have same eigenvalues.
linear-algebra eigenvalues-eigenvectors operator-theory
asked Jan 29 at 1:11
Alex TreadAlex Tread
132
$endgroup$
add a comment |
$begingroup$
I have this example on a textbook but it does not seem to be right.
Example:
If A,B are unitary operators and AB=BA,prove that A and B have same eigenvalues.
linear-algebra eigenvalues-eigenvectors operator-theory
asked Jan 29 at 1:11
Alex TreadAlex Tread
132
$endgroup$
add a comment |
$begingroup$
I have this example on a textbook but it does not seem to be right.
Example:
If A,B are unitary operators and AB=BA,prove that A and B have same eigenvalues.
linear-algebra eigenvalues-eigenvectors operator-theory
asked Jan 29 at 1:11
Alex TreadAlex Tread
132
$endgroup$
I have this example on a textbook but it does not seem to be right.
Example:
If A,B are unitary operators and AB=BA,prove that A and B have same eigenvalues.
linear-algebra eigenvalues-eigenvectors operator-theory
linear-algebra eigenvalues-eigenvectors operator-theory
asked Jan 29 at 1:11
Alex TreadAlex Tread
132
asked Jan 29 at 1:11
Alex TreadAlex Tread
132
edited Jan 29 at 3:04
user635162
asked Jan 29 at 1:11
Alex TreadAlex Tread
132
asked Jan 29 at 1:11
Alex TreadAlex Tread
132
asked Jan 29 at 1:11
Alex TreadAlex Tread
132
132
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
They do not necessarily have the same eigenvalues. Consider the operators $I$ and $-I$, which definitely commute, and are both self-adjoint and square to $I$. However, the eigenvalues of $I$ are just $1$, whereas the eigenvalues of $-I$ are just $-1$. You might be looking for the statement that commuting diagonalizable operators are simultaneously diagonalizable; that is, that they have the same eigenvectors.
edited Jan 29 at 7:04
user334732
4,33211240
answered Jan 29 at 1:23
Ashwin TrisalAshwin Trisal
1,2891516
$endgroup$
$begingroup$
Thanks, i tried to prove that they have same eigenvalues because i didn't thought it would be a mistake in textbook.
$endgroup$
– Alex Tread
Jan 29 at 10:43
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%2f3091620%2fdo-two-unitary-operators-that-commute-have-same-eigenvalues%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
$begingroup$
They do not necessarily have the same eigenvalues. Consider the operators $I$ and $-I$, which definitely commute, and are both self-adjoint and square to $I$. However, the eigenvalues of $I$ are just $1$, whereas the eigenvalues of $-I$ are just $-1$. You might be looking for the statement that commuting diagonalizable operators are simultaneously diagonalizable; that is, that they have the same eigenvectors.
edited Jan 29 at 7:04
user334732
4,33211240
answered Jan 29 at 1:23
Ashwin TrisalAshwin Trisal
1,2891516
$endgroup$
$begingroup$
Thanks, i tried to prove that they have same eigenvalues because i didn't thought it would be a mistake in textbook.
$endgroup$
– Alex Tread
Jan 29 at 10:43
add a comment |
$begingroup$
They do not necessarily have the same eigenvalues. Consider the operators $I$ and $-I$, which definitely commute, and are both self-adjoint and square to $I$. However, the eigenvalues of $I$ are just $1$, whereas the eigenvalues of $-I$ are just $-1$. You might be looking for the statement that commuting diagonalizable operators are simultaneously diagonalizable; that is, that they have the same eigenvectors.
edited Jan 29 at 7:04
user334732
4,33211240
answered Jan 29 at 1:23
Ashwin TrisalAshwin Trisal
1,2891516
$endgroup$
$begingroup$
Thanks, i tried to prove that they have same eigenvalues because i didn't thought it would be a mistake in textbook.
$endgroup$
– Alex Tread
Jan 29 at 10:43
add a comment |
$begingroup$
They do not necessarily have the same eigenvalues. Consider the operators $I$ and $-I$, which definitely commute, and are both self-adjoint and square to $I$. However, the eigenvalues of $I$ are just $1$, whereas the eigenvalues of $-I$ are just $-1$. You might be looking for the statement that commuting diagonalizable operators are simultaneously diagonalizable; that is, that they have the same eigenvectors.
edited Jan 29 at 7:04
user334732
4,33211240
answered Jan 29 at 1:23
Ashwin TrisalAshwin Trisal
1,2891516
$endgroup$
They do not necessarily have the same eigenvalues. Consider the operators $I$ and $-I$, which definitely commute, and are both self-adjoint and square to $I$. However, the eigenvalues of $I$ are just $1$, whereas the eigenvalues of $-I$ are just $-1$. You might be looking for the statement that commuting diagonalizable operators are simultaneously diagonalizable; that is, that they have the same eigenvectors.
edited Jan 29 at 7:04
user334732
4,33211240
answered Jan 29 at 1:23
Ashwin TrisalAshwin Trisal
1,2891516
edited Jan 29 at 7:04
user334732
4,33211240
edited Jan 29 at 7:04
user334732
4,33211240
edited Jan 29 at 7:04
user334732
4,33211240
4,33211240
answered Jan 29 at 1:23
Ashwin TrisalAshwin Trisal
1,2891516
answered Jan 29 at 1:23
Ashwin TrisalAshwin Trisal
1,2891516
answered Jan 29 at 1:23
Ashwin TrisalAshwin Trisal
1,2891516
1,2891516
$begingroup$
Thanks, i tried to prove that they have same eigenvalues because i didn't thought it would be a mistake in textbook.
$endgroup$
– Alex Tread
Jan 29 at 10:43
add a comment |
$begingroup$
Thanks, i tried to prove that they have same eigenvalues because i didn't thought it would be a mistake in textbook.
$endgroup$
– Alex Tread
Jan 29 at 10:43
$begingroup$
Thanks, i tried to prove that they have same eigenvalues because i didn't thought it would be a mistake in textbook.
$endgroup$
– Alex Tread
Jan 29 at 10:43
$begingroup$
Thanks, i tried to prove that they have same eigenvalues because i didn't thought it would be a mistake in textbook.
$endgroup$
– Alex Tread
Jan 29 at 10:43
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%2f3091620%2fdo-two-unitary-operators-that-commute-have-same-eigenvalues%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
