Mixing
How can I prove that $mathbb{S}_{+}^n$ is a closed and convex set?
$begingroup$
$mathbb{S}_{+}^n$ is the set of positive semidefinite (and symmetric) real matrices of size $ntimes n$. I have to prove that this set is a closed convex cone. How can I do?
proof-verification convex-analysis symmetric-matrices convex-geometry
edited Jan 17 at 4:38
max_zorn
3,39061329
asked Jan 16 at 13:41
Elisabetta BraviElisabetta Bravi
32
$endgroup$
add a comment |
$begingroup$
$mathbb{S}_{+}^n$ is the set of positive semidefinite (and symmetric) real matrices of size $ntimes n$. I have to prove that this set is a closed convex cone. How can I do?
proof-verification convex-analysis symmetric-matrices convex-geometry
edited Jan 17 at 4:38
max_zorn
3,39061329
asked Jan 16 at 13:41
Elisabetta BraviElisabetta Bravi
32
$endgroup$
2
$begingroup$
Do you know the definitions of closed, convex, and cone? Can you show any of the three properties?
$endgroup$
– Mees de Vries
Jan 16 at 13:49
$begingroup$
It is not closed. ${ 1 over n} I to 0$. The other properties are just a matter of grinding through the definitions.
$endgroup$
– copper.hat
Jan 16 at 15:05
$begingroup$
I expect positive means positive definite as in $Axcdot xgeq0$ for all $x$.
$endgroup$
– SmileyCraft
Jan 16 at 15:18
$begingroup$
@SmileyCraft that would be positive semidefinite
$endgroup$
– LinAlg
Jan 16 at 15:31
add a comment |
$begingroup$
$mathbb{S}_{+}^n$ is the set of positive semidefinite (and symmetric) real matrices of size $ntimes n$. I have to prove that this set is a closed convex cone. How can I do?
proof-verification convex-analysis symmetric-matrices convex-geometry
edited Jan 17 at 4:38
max_zorn
3,39061329
asked Jan 16 at 13:41
Elisabetta BraviElisabetta Bravi
32
$endgroup$
$mathbb{S}_{+}^n$ is the set of positive semidefinite (and symmetric) real matrices of size $ntimes n$. I have to prove that this set is a closed convex cone. How can I do?
proof-verification convex-analysis symmetric-matrices convex-geometry
proof-verification convex-analysis symmetric-matrices convex-geometry
edited Jan 17 at 4:38
max_zorn
3,39061329
asked Jan 16 at 13:41
Elisabetta BraviElisabetta Bravi
32
edited Jan 17 at 4:38
max_zorn
3,39061329
asked Jan 16 at 13:41
Elisabetta BraviElisabetta Bravi
32
edited Jan 17 at 4:38
max_zorn
3,39061329
edited Jan 17 at 4:38
max_zorn
3,39061329
edited Jan 17 at 4:38
max_zorn
3,39061329
3,39061329
asked Jan 16 at 13:41
Elisabetta BraviElisabetta Bravi
32
asked Jan 16 at 13:41
Elisabetta BraviElisabetta Bravi
32
asked Jan 16 at 13:41
Elisabetta BraviElisabetta Bravi
32
32
2
$begingroup$
Do you know the definitions of closed, convex, and cone? Can you show any of the three properties?
$endgroup$
– Mees de Vries
Jan 16 at 13:49
$begingroup$
It is not closed. ${ 1 over n} I to 0$. The other properties are just a matter of grinding through the definitions.
$endgroup$
– copper.hat
Jan 16 at 15:05
$begingroup$
I expect positive means positive definite as in $Axcdot xgeq0$ for all $x$.
$endgroup$
– SmileyCraft
Jan 16 at 15:18
$begingroup$
@SmileyCraft that would be positive semidefinite
$endgroup$
– LinAlg
Jan 16 at 15:31
add a comment |
2
$begingroup$
Do you know the definitions of closed, convex, and cone? Can you show any of the three properties?
$endgroup$
– Mees de Vries
Jan 16 at 13:49
$begingroup$
It is not closed. ${ 1 over n} I to 0$. The other properties are just a matter of grinding through the definitions.
$endgroup$
– copper.hat
Jan 16 at 15:05
$begingroup$
I expect positive means positive definite as in $Axcdot xgeq0$ for all $x$.
$endgroup$
– SmileyCraft
Jan 16 at 15:18
$begingroup$
@SmileyCraft that would be positive semidefinite
$endgroup$
– LinAlg
Jan 16 at 15:31
2
2
$begingroup$
Do you know the definitions of closed, convex, and cone? Can you show any of the three properties?
$endgroup$
– Mees de Vries
Jan 16 at 13:49
$begingroup$
Do you know the definitions of closed, convex, and cone? Can you show any of the three properties?
$endgroup$
– Mees de Vries
Jan 16 at 13:49
$begingroup$
It is not closed. ${ 1 over n} I to 0$. The other properties are just a matter of grinding through the definitions.
$endgroup$
– copper.hat
Jan 16 at 15:05
$begingroup$
It is not closed. ${ 1 over n} I to 0$. The other properties are just a matter of grinding through the definitions.
$endgroup$
– copper.hat
Jan 16 at 15:05
$begingroup$
I expect positive means positive definite as in $Axcdot xgeq0$ for all $x$.
$endgroup$
– SmileyCraft
Jan 16 at 15:18
$begingroup$
I expect positive means positive definite as in $Axcdot xgeq0$ for all $x$.
$endgroup$
– SmileyCraft
Jan 16 at 15:18
$begingroup$
@SmileyCraft that would be positive semidefinite
$endgroup$
– LinAlg
Jan 16 at 15:31
$begingroup$
@SmileyCraft that would be positive semidefinite
$endgroup$
– LinAlg
Jan 16 at 15:31
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Note that $A ge 0$ iff $x^TAx ge 0$ for all $x$ iff $A in cap_x { B | x^T B x
ge 0}$.
Note that for any $x$ that ${ B | x^T B x ge 0}$ is a closed half space (hence convex). Since $0 in { B | x^T B x ge 0}$ we see that it is a cone as well.
answered Jan 17 at 18:50
copper.hatcopper.hat
127k559160
$endgroup$
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%2f3075742%2fhow-can-i-prove-that-mathbbs-n-is-a-closed-and-convex-set%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
$begingroup$
Note that $A ge 0$ iff $x^TAx ge 0$ for all $x$ iff $A in cap_x { B | x^T B x
ge 0}$.
Note that for any $x$ that ${ B | x^T B x ge 0}$ is a closed half space (hence convex). Since $0 in { B | x^T B x ge 0}$ we see that it is a cone as well.
answered Jan 17 at 18:50
copper.hatcopper.hat
127k559160
$endgroup$
add a comment |
$begingroup$
Note that $A ge 0$ iff $x^TAx ge 0$ for all $x$ iff $A in cap_x { B | x^T B x
ge 0}$.
Note that for any $x$ that ${ B | x^T B x ge 0}$ is a closed half space (hence convex). Since $0 in { B | x^T B x ge 0}$ we see that it is a cone as well.
answered Jan 17 at 18:50
copper.hatcopper.hat
127k559160
$endgroup$
add a comment |
$begingroup$
Note that $A ge 0$ iff $x^TAx ge 0$ for all $x$ iff $A in cap_x { B | x^T B x
ge 0}$.
Note that for any $x$ that ${ B | x^T B x ge 0}$ is a closed half space (hence convex). Since $0 in { B | x^T B x ge 0}$ we see that it is a cone as well.
answered Jan 17 at 18:50
copper.hatcopper.hat
127k559160
$endgroup$
Note that $A ge 0$ iff $x^TAx ge 0$ for all $x$ iff $A in cap_x { B | x^T B x
ge 0}$.
Note that for any $x$ that ${ B | x^T B x ge 0}$ is a closed half space (hence convex). Since $0 in { B | x^T B x ge 0}$ we see that it is a cone as well.
answered Jan 17 at 18:50
copper.hatcopper.hat
127k559160
edited Jan 17 at 23:07
answered Jan 17 at 18:50
copper.hatcopper.hat
127k559160
answered Jan 17 at 18:50
copper.hatcopper.hat
127k559160
answered Jan 17 at 18:50
copper.hatcopper.hat
127k559160
127k559160
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%2f3075742%2fhow-can-i-prove-that-mathbbs-n-is-a-closed-and-convex-set%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
$begingroup$
Do you know the definitions of closed, convex, and cone? Can you show any of the three properties?
$endgroup$
– Mees de Vries
Jan 16 at 13:49
$begingroup$
It is not closed. ${ 1 over n} I to 0$. The other properties are just a matter of grinding through the definitions.
$endgroup$
– copper.hat
Jan 16 at 15:05
$begingroup$
I expect positive means positive definite as in $Axcdot xgeq0$ for all $x$.
$endgroup$
– SmileyCraft
Jan 16 at 15:18
$begingroup$
@SmileyCraft that would be positive semidefinite
$endgroup$
– LinAlg
Jan 16 at 15:31