Mixing
Why can a constraint on a matrix being positive definite be rewritten as the matrix minus the identity being...
$begingroup$
My instructor today mentioned that if we have a constraint that a matrix $A$ is positive definite, then we can rewrite this constraint as $A - I$ is positive semidefinite without this affecting the feasibility of the problem.
Why wouldn't this affect the feasibility? The eigenvalues of $A - I$ would be one less than all the eigenvalues of A, so if A has an eigenvalue = $1/2$, wouldn't $A - I$ have an eigenvalue that is $-1/2$, changing the feasibility of the problem?
eigenvalues-eigenvectors positive-definite positive-semidefinite semidefinite-programming
asked Jan 17 at 21:15
Ronald GRonald G
61
$endgroup$
add a comment |
$begingroup$
My instructor today mentioned that if we have a constraint that a matrix $A$ is positive definite, then we can rewrite this constraint as $A - I$ is positive semidefinite without this affecting the feasibility of the problem.
Why wouldn't this affect the feasibility? The eigenvalues of $A - I$ would be one less than all the eigenvalues of A, so if A has an eigenvalue = $1/2$, wouldn't $A - I$ have an eigenvalue that is $-1/2$, changing the feasibility of the problem?
eigenvalues-eigenvectors positive-definite positive-semidefinite semidefinite-programming
asked Jan 17 at 21:15
Ronald GRonald G
61
$endgroup$
2
$begingroup$
should be $A- varepsilon I$
$endgroup$
– Will Jagy
Jan 17 at 21:21
add a comment |
$begingroup$
My instructor today mentioned that if we have a constraint that a matrix $A$ is positive definite, then we can rewrite this constraint as $A - I$ is positive semidefinite without this affecting the feasibility of the problem.
Why wouldn't this affect the feasibility? The eigenvalues of $A - I$ would be one less than all the eigenvalues of A, so if A has an eigenvalue = $1/2$, wouldn't $A - I$ have an eigenvalue that is $-1/2$, changing the feasibility of the problem?
eigenvalues-eigenvectors positive-definite positive-semidefinite semidefinite-programming
asked Jan 17 at 21:15
Ronald GRonald G
61
$endgroup$
My instructor today mentioned that if we have a constraint that a matrix $A$ is positive definite, then we can rewrite this constraint as $A - I$ is positive semidefinite without this affecting the feasibility of the problem.
Why wouldn't this affect the feasibility? The eigenvalues of $A - I$ would be one less than all the eigenvalues of A, so if A has an eigenvalue = $1/2$, wouldn't $A - I$ have an eigenvalue that is $-1/2$, changing the feasibility of the problem?
eigenvalues-eigenvectors positive-definite positive-semidefinite semidefinite-programming
eigenvalues-eigenvectors positive-definite positive-semidefinite semidefinite-programming
asked Jan 17 at 21:15
Ronald GRonald G
61
asked Jan 17 at 21:15
Ronald GRonald G
61
asked Jan 17 at 21:15
Ronald GRonald G
61
asked Jan 17 at 21:15
Ronald GRonald G
61
asked Jan 17 at 21:15
Ronald GRonald G
61
61
2
$begingroup$
should be $A- varepsilon I$
$endgroup$
– Will Jagy
Jan 17 at 21:21
add a comment |
2
$begingroup$
should be $A- varepsilon I$
$endgroup$
– Will Jagy
Jan 17 at 21:21
2
2
$begingroup$
should be $A- varepsilon I$
$endgroup$
– Will Jagy
Jan 17 at 21:21
$begingroup$
should be $A- varepsilon I$
$endgroup$
– Will Jagy
Jan 17 at 21:21
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%2f3077516%2fwhy-can-a-constraint-on-a-matrix-being-positive-definite-be-rewritten-as-the-mat%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%2f3077516%2fwhy-can-a-constraint-on-a-matrix-being-positive-definite-be-rewritten-as-the-mat%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
2
$begingroup$
should be $A- varepsilon I$
$endgroup$
– Will Jagy
Jan 17 at 21:21