Whether the set of a symmetric copositive matrices $x^T A x geq 0$ for $forall x in mathbbR^n$ $x geq 0$ is convex? The Next CEO of Stack OverflowNotation for the set of symmetric matrices and symmetric positive definite matricesIs this set of matrices convex?How to prove that the set of real $ n times n $ PSD matrices of rank $leq r < n $ and unit trace, is not contractible?Convex hull of rotation matrices is closed and contains the originIs the the set of symmetric positive-definite matrices in the space of all symmetric matrices strictly convex?Given $P$ a real symmetric positive semidefinite matrix, what matrix $A$ ensure that $A^TPA$ is PSD?Use the definition of convexity to show that $A$ is a convex set.Show that a set of positive semidefinite (PSD) matrices is a convex setProjection of a Symmetric Matrix onto the space of Positive Semi-Definite MatricesHow can I prove that $mathbbS_+^n$ is a closed and convex set?
Implement the Thanos sorting algorithm
Why Were Madagascar and New Zealand Discovered So Late?
Return the Closest Prime Number
Is it okay to store user locations?
Too much space between section and text in a twocolumn document
How to start emacs in "nothing" mode (`fundamental-mode`)
If the heap is initialized for security, then why is the stack uninitialized?
How long to clear the 'suck zone' of a turbofan after start is initiated?
When airplanes disconnect from a tanker during air to air refueling, why do they bank so sharply to the right?
How to write the block matrix in LaTex?
% symbol leads to superlong (forever?) compilations
How do I solve this limit?
Anatomically Correct Strange Women In Ponds Distributing Swords
How to use tikz in fbox?
Does it take more energy to get to Venus or to Mars?
Why did we only see the N-1 starfighters in one film?
If I blow insulation everywhere in my attic except the door trap, will heat escape through it?
What does "Its cash flow is deeply negative" mean?
Opposite of a diet
What does this shorthand mean?
What is the purpose of the Evocation wizard's Potent Cantrip feature?
Rotate a column
Can a single photon have an energy density?
What happens if you roll doubles 3 times then land on "Go to jail?"
Whether the set of a symmetric copositive matrices $x^T A x geq 0$ for $forall x in mathbbR^n$ $x geq 0$ is convex?
The Next CEO of Stack OverflowNotation for the set of symmetric matrices and symmetric positive definite matricesIs this set of matrices convex?How to prove that the set of real $ n times n $ PSD matrices of rank $leq r < n $ and unit trace, is not contractible?Convex hull of rotation matrices is closed and contains the originIs the the set of symmetric positive-definite matrices in the space of all symmetric matrices strictly convex?Given $P$ a real symmetric positive semidefinite matrix, what matrix $A$ ensure that $A^TPA$ is PSD?Use the definition of convexity to show that $A$ is a convex set.Show that a set of positive semidefinite (PSD) matrices is a convex setProjection of a Symmetric Matrix onto the space of Positive Semi-Definite MatricesHow can I prove that $mathbbS_+^n$ is a closed and convex set?
$begingroup$
Whether the set of a $n times n$ symmetric copositive matrices $x^T A x geq 0$ $forall x in mathbbR^n$ and $x geq 0$ is convex?
Attempt:
If I show that the set of the positive semidefinite (psd) matrices is convex, then the intersection of the set of psd matrices and $x geq 0$ is also convex. Is that correct?
matrices convex-analysis copositivity
edited yesterday
Rodrigo de Azevedo
13.2k41960
asked yesterday
learninglearning
617
$endgroup$
add a comment |
$begingroup$
Whether the set of a $n times n$ symmetric copositive matrices $x^T A x geq 0$ $forall x in mathbbR^n$ and $x geq 0$ is convex?
Attempt:
If I show that the set of the positive semidefinite (psd) matrices is convex, then the intersection of the set of psd matrices and $x geq 0$ is also convex. Is that correct?
matrices convex-analysis copositivity
edited yesterday
Rodrigo de Azevedo
13.2k41960
asked yesterday
learninglearning
617
$endgroup$
$begingroup$
What is "copositive" ?
$endgroup$
– kimchi lover
yesterday
$begingroup$
@kimchilover en.wikipedia.org/wiki/Copositive_matrix
$endgroup$
– Yanko
yesterday
add a comment |
$begingroup$
Whether the set of a $n times n$ symmetric copositive matrices $x^T A x geq 0$ $forall x in mathbbR^n$ and $x geq 0$ is convex?
Attempt:
If I show that the set of the positive semidefinite (psd) matrices is convex, then the intersection of the set of psd matrices and $x geq 0$ is also convex. Is that correct?
matrices convex-analysis copositivity
edited yesterday
Rodrigo de Azevedo
13.2k41960
asked yesterday
learninglearning
617
$endgroup$
Whether the set of a $n times n$ symmetric copositive matrices $x^T A x geq 0$ $forall x in mathbbR^n$ and $x geq 0$ is convex?
Attempt:
If I show that the set of the positive semidefinite (psd) matrices is convex, then the intersection of the set of psd matrices and $x geq 0$ is also convex. Is that correct?
matrices convex-analysis copositivity
matrices convex-analysis copositivity
edited yesterday
Rodrigo de Azevedo
13.2k41960
asked yesterday
learninglearning
617
edited yesterday
Rodrigo de Azevedo
13.2k41960
asked yesterday
learninglearning
617
edited yesterday
Rodrigo de Azevedo
13.2k41960
edited yesterday
Rodrigo de Azevedo
13.2k41960
edited yesterday
Rodrigo de Azevedo
13.2k41960
13.2k41960
asked yesterday
learninglearning
617
asked yesterday
learninglearning
617
asked yesterday
learninglearning
617
617
$begingroup$
What is "copositive" ?
$endgroup$
– kimchi lover
yesterday
$begingroup$
@kimchilover en.wikipedia.org/wiki/Copositive_matrix
$endgroup$
– Yanko
yesterday
add a comment |
$begingroup$
What is "copositive" ?
$endgroup$
– kimchi lover
yesterday
$begingroup$
@kimchilover en.wikipedia.org/wiki/Copositive_matrix
$endgroup$
– Yanko
yesterday
$begingroup$
What is "copositive" ?
$endgroup$
– kimchi lover
yesterday
$begingroup$
What is "copositive" ?
$endgroup$
– kimchi lover
yesterday
$begingroup$
@kimchilover en.wikipedia.org/wiki/Copositive_matrix
$endgroup$
– Yanko
yesterday
$begingroup$
@kimchilover en.wikipedia.org/wiki/Copositive_matrix
$endgroup$
– Yanko
yesterday
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
No need to do that! Note that if $A,B$ are copositive, then for any $0<lambda<1$ and $xge 0$:$$x^TBigg(lambda A+(1-lambda)BBigg)x=lambda x^TAx+(1-lambda )x^TBxge 0$$which implies that also $lambda A+(1-lambda)B$ is copositive and hence the set of copositive matrices is convex.
answered yesterday
Mostafa AyazMostafa Ayaz
18.1k31040
$endgroup$
add a comment |
$begingroup$
You can follow these steps.
Let $mathcalF$ denote the family of all the set of a $ntimes n$ symmetric copositive matrices.
Now prove the following two assertions: The first one is,
For any $lambda>0$, $AinmathcalF$ implies $lambda AinmathcalA$.
Which follows from the fact that $x^T (lambda A) x = lambda (x^T Ax)$.
The second is,
For any $A,BinmathcalF$, $A+BinmathcalF$.
Which you can do by calculation $x^T (A+B)x$.
Once done. You can combine those two to show that
For any $A,BinmathcalF$ and $lambdain[0,1]$ we have $lambda A + (1-lambda) B inmathcalF$
which means, by definition, that $mathcalF$ is convex.
answered yesterday
YankoYanko
8,1282830
$endgroup$
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%2f3164334%2fwhether-the-set-of-a-symmetric-copositive-matrices-xt-a-x-geq-0-for-forall%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
$begingroup$
No need to do that! Note that if $A,B$ are copositive, then for any $0<lambda<1$ and $xge 0$:$$x^TBigg(lambda A+(1-lambda)BBigg)x=lambda x^TAx+(1-lambda )x^TBxge 0$$which implies that also $lambda A+(1-lambda)B$ is copositive and hence the set of copositive matrices is convex.
answered yesterday
Mostafa AyazMostafa Ayaz
18.1k31040
$endgroup$
add a comment |
$begingroup$
No need to do that! Note that if $A,B$ are copositive, then for any $0<lambda<1$ and $xge 0$:$$x^TBigg(lambda A+(1-lambda)BBigg)x=lambda x^TAx+(1-lambda )x^TBxge 0$$which implies that also $lambda A+(1-lambda)B$ is copositive and hence the set of copositive matrices is convex.
answered yesterday
Mostafa AyazMostafa Ayaz
18.1k31040
$endgroup$
add a comment |
$begingroup$
No need to do that! Note that if $A,B$ are copositive, then for any $0<lambda<1$ and $xge 0$:$$x^TBigg(lambda A+(1-lambda)BBigg)x=lambda x^TAx+(1-lambda )x^TBxge 0$$which implies that also $lambda A+(1-lambda)B$ is copositive and hence the set of copositive matrices is convex.
answered yesterday
Mostafa AyazMostafa Ayaz
18.1k31040
$endgroup$
No need to do that! Note that if $A,B$ are copositive, then for any $0<lambda<1$ and $xge 0$:$$x^TBigg(lambda A+(1-lambda)BBigg)x=lambda x^TAx+(1-lambda )x^TBxge 0$$which implies that also $lambda A+(1-lambda)B$ is copositive and hence the set of copositive matrices is convex.
answered yesterday
Mostafa AyazMostafa Ayaz
18.1k31040
answered yesterday
Mostafa AyazMostafa Ayaz
18.1k31040
answered yesterday
Mostafa AyazMostafa Ayaz
18.1k31040
answered yesterday
Mostafa AyazMostafa Ayaz
18.1k31040
18.1k31040
add a comment |
add a comment |
$begingroup$
You can follow these steps.
Let $mathcalF$ denote the family of all the set of a $ntimes n$ symmetric copositive matrices.
Now prove the following two assertions: The first one is,
For any $lambda>0$, $AinmathcalF$ implies $lambda AinmathcalA$.
Which follows from the fact that $x^T (lambda A) x = lambda (x^T Ax)$.
The second is,
For any $A,BinmathcalF$, $A+BinmathcalF$.
Which you can do by calculation $x^T (A+B)x$.
Once done. You can combine those two to show that
For any $A,BinmathcalF$ and $lambdain[0,1]$ we have $lambda A + (1-lambda) B inmathcalF$
which means, by definition, that $mathcalF$ is convex.
answered yesterday
YankoYanko
8,1282830
$endgroup$
add a comment |
$begingroup$
You can follow these steps.
Let $mathcalF$ denote the family of all the set of a $ntimes n$ symmetric copositive matrices.
Now prove the following two assertions: The first one is,
For any $lambda>0$, $AinmathcalF$ implies $lambda AinmathcalA$.
Which follows from the fact that $x^T (lambda A) x = lambda (x^T Ax)$.
The second is,
For any $A,BinmathcalF$, $A+BinmathcalF$.
Which you can do by calculation $x^T (A+B)x$.
Once done. You can combine those two to show that
For any $A,BinmathcalF$ and $lambdain[0,1]$ we have $lambda A + (1-lambda) B inmathcalF$
which means, by definition, that $mathcalF$ is convex.
answered yesterday
YankoYanko
8,1282830
$endgroup$
add a comment |
$begingroup$
You can follow these steps.
Let $mathcalF$ denote the family of all the set of a $ntimes n$ symmetric copositive matrices.
Now prove the following two assertions: The first one is,
For any $lambda>0$, $AinmathcalF$ implies $lambda AinmathcalA$.
Which follows from the fact that $x^T (lambda A) x = lambda (x^T Ax)$.
The second is,
For any $A,BinmathcalF$, $A+BinmathcalF$.
Which you can do by calculation $x^T (A+B)x$.
Once done. You can combine those two to show that
For any $A,BinmathcalF$ and $lambdain[0,1]$ we have $lambda A + (1-lambda) B inmathcalF$
which means, by definition, that $mathcalF$ is convex.
answered yesterday
YankoYanko
8,1282830
$endgroup$
You can follow these steps.
Let $mathcalF$ denote the family of all the set of a $ntimes n$ symmetric copositive matrices.
Now prove the following two assertions: The first one is,
For any $lambda>0$, $AinmathcalF$ implies $lambda AinmathcalA$.
Which follows from the fact that $x^T (lambda A) x = lambda (x^T Ax)$.
The second is,
For any $A,BinmathcalF$, $A+BinmathcalF$.
Which you can do by calculation $x^T (A+B)x$.
Once done. You can combine those two to show that
For any $A,BinmathcalF$ and $lambdain[0,1]$ we have $lambda A + (1-lambda) B inmathcalF$
which means, by definition, that $mathcalF$ is convex.
answered yesterday
YankoYanko
8,1282830
answered yesterday
YankoYanko
8,1282830
answered yesterday
YankoYanko
8,1282830
answered yesterday
YankoYanko
8,1282830
8,1282830
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%2f3164334%2fwhether-the-set-of-a-symmetric-copositive-matrices-xt-a-x-geq-0-for-forall%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
$begingroup$
What is "copositive" ?
$endgroup$
– kimchi lover
yesterday
$begingroup$
@kimchilover en.wikipedia.org/wiki/Copositive_matrix
$endgroup$
– Yanko
yesterday