Weakly Diagonally Dominant with Positive Diagonals
$begingroup$
Suppose $A in mathbb{R}^{ntimes n}$ (not necessarily symmetric) is weakly diagonally dominant with positive diagonals. Is it true that all eigenvalues of $A$ are non-negative (in the case of complex eigenvalues, have non-negative real parts)? Note that this holds true for strictly diagonally dominant matrices: if the matrix is symmetric with non-negative diagonal entries, the matrix is positive semi-definite.
linear-algebra matrices matrix-calculus diagonalization
asked Dec 12 '18 at 4:30
ArthurArthur
48512
$endgroup$
add a comment |
$begingroup$
Suppose $A in mathbb{R}^{ntimes n}$ (not necessarily symmetric) is weakly diagonally dominant with positive diagonals. Is it true that all eigenvalues of $A$ are non-negative (in the case of complex eigenvalues, have non-negative real parts)? Note that this holds true for strictly diagonally dominant matrices: if the matrix is symmetric with non-negative diagonal entries, the matrix is positive semi-definite.
linear-algebra matrices matrix-calculus diagonalization
asked Dec 12 '18 at 4:30
ArthurArthur
48512
$endgroup$
$begingroup$
Correct! But, in the case of complex eigenvalues, we need the real part of the eigenvalues to be non-negative.
$endgroup$
– Arthur
Dec 12 '18 at 4:55
$begingroup$
Are you familiar with the Gershgorin circle theorem?
$endgroup$
– JimmyK4542
Dec 12 '18 at 4:59
$begingroup$
Yes! I am familiar with that.
$endgroup$
– Arthur
Dec 12 '18 at 5:00
$begingroup$
I guess I got your point. In this case, all circles in the Gershgorin circle theorem will be on the right-half plane will possible intersection with the imaginary axis!
$endgroup$
– Arthur
Dec 12 '18 at 5:05
add a comment |
$begingroup$
Suppose $A in mathbb{R}^{ntimes n}$ (not necessarily symmetric) is weakly diagonally dominant with positive diagonals. Is it true that all eigenvalues of $A$ are non-negative (in the case of complex eigenvalues, have non-negative real parts)? Note that this holds true for strictly diagonally dominant matrices: if the matrix is symmetric with non-negative diagonal entries, the matrix is positive semi-definite.
linear-algebra matrices matrix-calculus diagonalization
asked Dec 12 '18 at 4:30
ArthurArthur
48512
$endgroup$
Suppose $A in mathbb{R}^{ntimes n}$ (not necessarily symmetric) is weakly diagonally dominant with positive diagonals. Is it true that all eigenvalues of $A$ are non-negative (in the case of complex eigenvalues, have non-negative real parts)? Note that this holds true for strictly diagonally dominant matrices: if the matrix is symmetric with non-negative diagonal entries, the matrix is positive semi-definite.
linear-algebra matrices matrix-calculus diagonalization
linear-algebra matrices matrix-calculus diagonalization
asked Dec 12 '18 at 4:30
ArthurArthur
48512
asked Dec 12 '18 at 4:30
ArthurArthur
48512
edited Dec 12 '18 at 4:58
Arthur
asked Dec 12 '18 at 4:30
ArthurArthur
48512
asked Dec 12 '18 at 4:30
ArthurArthur
48512
asked Dec 12 '18 at 4:30
ArthurArthur
48512
48512
$begingroup$
Correct! But, in the case of complex eigenvalues, we need the real part of the eigenvalues to be non-negative.
$endgroup$
– Arthur
Dec 12 '18 at 4:55
$begingroup$
Are you familiar with the Gershgorin circle theorem?
$endgroup$
– JimmyK4542
Dec 12 '18 at 4:59
$begingroup$
Yes! I am familiar with that.
$endgroup$
– Arthur
Dec 12 '18 at 5:00
$begingroup$
I guess I got your point. In this case, all circles in the Gershgorin circle theorem will be on the right-half plane will possible intersection with the imaginary axis!
$endgroup$
– Arthur
Dec 12 '18 at 5:05
add a comment |
$begingroup$
Correct! But, in the case of complex eigenvalues, we need the real part of the eigenvalues to be non-negative.
$endgroup$
– Arthur
Dec 12 '18 at 4:55
$begingroup$
Are you familiar with the Gershgorin circle theorem?
$endgroup$
– JimmyK4542
Dec 12 '18 at 4:59
$begingroup$
Yes! I am familiar with that.
$endgroup$
– Arthur
Dec 12 '18 at 5:00
$begingroup$
I guess I got your point. In this case, all circles in the Gershgorin circle theorem will be on the right-half plane will possible intersection with the imaginary axis!
$endgroup$
– Arthur
Dec 12 '18 at 5:05
$begingroup$
Correct! But, in the case of complex eigenvalues, we need the real part of the eigenvalues to be non-negative.
$endgroup$
– Arthur
Dec 12 '18 at 4:55
$begingroup$
Correct! But, in the case of complex eigenvalues, we need the real part of the eigenvalues to be non-negative.
$endgroup$
– Arthur
Dec 12 '18 at 4:55
$begingroup$
Are you familiar with the Gershgorin circle theorem?
$endgroup$
– JimmyK4542
Dec 12 '18 at 4:59
$begingroup$
Are you familiar with the Gershgorin circle theorem?
$endgroup$
– JimmyK4542
Dec 12 '18 at 4:59
$begingroup$
Yes! I am familiar with that.
$endgroup$
– Arthur
Dec 12 '18 at 5:00
$begingroup$
Yes! I am familiar with that.
$endgroup$
– Arthur
Dec 12 '18 at 5:00
$begingroup$
I guess I got your point. In this case, all circles in the Gershgorin circle theorem will be on the right-half plane will possible intersection with the imaginary axis!
$endgroup$
– Arthur
Dec 12 '18 at 5:05
$begingroup$
I guess I got your point. In this case, all circles in the Gershgorin circle theorem will be on the right-half plane will possible intersection with the imaginary axis!
$endgroup$
– Arthur
Dec 12 '18 at 5:05
add a comment |
0
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%2f3036228%2fweakly-diagonally-dominant-with-positive-diagonals%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
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.
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%2f3036228%2fweakly-diagonally-dominant-with-positive-diagonals%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
$begingroup$
Correct! But, in the case of complex eigenvalues, we need the real part of the eigenvalues to be non-negative.
$endgroup$
– Arthur
Dec 12 '18 at 4:55
$begingroup$
Are you familiar with the Gershgorin circle theorem?
$endgroup$
– JimmyK4542
Dec 12 '18 at 4:59
$begingroup$
Yes! I am familiar with that.
$endgroup$
– Arthur
Dec 12 '18 at 5:00
$begingroup$
I guess I got your point. In this case, all circles in the Gershgorin circle theorem will be on the right-half plane will possible intersection with the imaginary axis!
$endgroup$
– Arthur
Dec 12 '18 at 5:05