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 singularvalues
asked Dec 3 '18 at 15:12
ITAITA
1,021620
$endgroup$
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 singularvalues
asked Dec 3 '18 at 15:12
ITAITA
1,021620
$endgroup$
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 singularvalues
asked Dec 3 '18 at 15:12
ITAITA
1,021620
$endgroup$
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 singularvalues
linear-algebra singularvalues
asked Dec 3 '18 at 15:12
ITAITA
1,021620
asked Dec 3 '18 at 15:12
ITAITA
1,021620
asked Dec 3 '18 at 15:12
ITAITA
1,021620
asked Dec 3 '18 at 15:12
ITAITA
1,021620
asked Dec 3 '18 at 15:12
ITAITA
1,021620
1,021620
add a comment |
add a comment |
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