Relationship between least singular value and exact cover of image
$begingroup$
Is the following fact true? If so, how would one go about providing a proof or if not then disproving?
The least singular value of a matrix determines exactly the radius of the largest ball that can precisely cover the image of the unit ball under the matrix action.
I have to make clear what I mean by "precisely cover".
In the following picture I have drawn an ellipse and put a circle along the semi-minor axis. Suppose the ellipse was the image of of the unit ball in $mathbb{R}^3$ via a matrix $A in mathbb{R}^{2 times 3}$. Then we know that the major and minor axis lengths are related to the smallest and largest singular values of $A$.
Then is it true that we can cover up all the points in the ellipse by moving this ball around in the ellipse without any part of the ball ever going outside the ellipse? Is it also true that this is the largest ball that does the job, for any larger radius will result in inaccessibility or rather force part of the ball to be outside the ellipse to obtain a covering? For example the picture below, some points can't be covered without having the ball be outside the ellipse.
linear-algebra general-topology singularvalues
asked Dec 3 '18 at 15:12
ITAITA
1,041621
$endgroup$
add a comment |
$begingroup$
Is the following fact true? If so, how would one go about providing a proof or if not then disproving?
The least singular value of a matrix determines exactly the radius of the largest ball that can precisely cover the image of the unit ball under the matrix action.
I have to make clear what I mean by "precisely cover".
In the following picture I have drawn an ellipse and put a circle along the semi-minor axis. Suppose the ellipse was the image of of the unit ball in $mathbb{R}^3$ via a matrix $A in mathbb{R}^{2 times 3}$. Then we know that the major and minor axis lengths are related to the smallest and largest singular values of $A$.
Then is it true that we can cover up all the points in the ellipse by moving this ball around in the ellipse without any part of the ball ever going outside the ellipse? Is it also true that this is the largest ball that does the job, for any larger radius will result in inaccessibility or rather force part of the ball to be outside the ellipse to obtain a covering? For example the picture below, some points can't be covered without having the ball be outside the ellipse.
linear-algebra general-topology singularvalues
asked Dec 3 '18 at 15:12
ITAITA
1,041621
$endgroup$
add a comment |
$begingroup$
Is the following fact true? If so, how would one go about providing a proof or if not then disproving?
The least singular value of a matrix determines exactly the radius of the largest ball that can precisely cover the image of the unit ball under the matrix action.
I have to make clear what I mean by "precisely cover".
In the following picture I have drawn an ellipse and put a circle along the semi-minor axis. Suppose the ellipse was the image of of the unit ball in $mathbb{R}^3$ via a matrix $A in mathbb{R}^{2 times 3}$. Then we know that the major and minor axis lengths are related to the smallest and largest singular values of $A$.
Then is it true that we can cover up all the points in the ellipse by moving this ball around in the ellipse without any part of the ball ever going outside the ellipse? Is it also true that this is the largest ball that does the job, for any larger radius will result in inaccessibility or rather force part of the ball to be outside the ellipse to obtain a covering? For example the picture below, some points can't be covered without having the ball be outside the ellipse.
linear-algebra general-topology singularvalues
asked Dec 3 '18 at 15:12
ITAITA
1,041621
$endgroup$
Is the following fact true? If so, how would one go about providing a proof or if not then disproving?
The least singular value of a matrix determines exactly the radius of the largest ball that can precisely cover the image of the unit ball under the matrix action.
I have to make clear what I mean by "precisely cover".
In the following picture I have drawn an ellipse and put a circle along the semi-minor axis. Suppose the ellipse was the image of of the unit ball in $mathbb{R}^3$ via a matrix $A in mathbb{R}^{2 times 3}$. Then we know that the major and minor axis lengths are related to the smallest and largest singular values of $A$.
Then is it true that we can cover up all the points in the ellipse by moving this ball around in the ellipse without any part of the ball ever going outside the ellipse? Is it also true that this is the largest ball that does the job, for any larger radius will result in inaccessibility or rather force part of the ball to be outside the ellipse to obtain a covering? For example the picture below, some points can't be covered without having the ball be outside the ellipse.
linear-algebra general-topology singularvalues
linear-algebra general-topology singularvalues
asked Dec 3 '18 at 15:12
ITAITA
1,041621
asked Dec 3 '18 at 15:12
ITAITA
1,041621
edited Feb 7 at 3:43
ITA
asked Dec 3 '18 at 15:12
ITAITA
1,041621
asked Dec 3 '18 at 15:12
ITAITA
1,041621
asked Dec 3 '18 at 15:12
ITAITA
1,041621
1,041621
add a comment |
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%2f3024184%2frelationship-between-least-singular-value-and-exact-cover-of-image%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%2f3024184%2frelationship-between-least-singular-value-and-exact-cover-of-image%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
