show singular matrix map interior and surface of a unit sphere into an ellipse The 2019 Stack Overflow Developer Survey Results Are InShowing a matrix can't be factored into unit lower triangular matrix and upper triangular matrixRE: singular matrix and eigenvectorsProve that $textrank(A) = textrank(A^T)$ using SVDRelationship between eigenvectors and singular vectors of a Hermitian matrix?Image of unit sphere being hyper ellipse proof (SVD)symmetric matrix right and left singular matrix of a symmetric matrixSingular value decomposition inequalityBest singular unit trace matrix approximationProve that rank(A) = rank(A$^dagger$)=rank(AA$^dagger$)=rank(A$^dagger$A) using the SVD decompositionSingular Value Decomposition of a Real Unit Matrix
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show singular matrix map interior and surface of a unit sphere into an ellipse
The 2019 Stack Overflow Developer Survey Results Are InShowing a matrix can't be factored into unit lower triangular matrix and upper triangular matrixRE: singular matrix and eigenvectorsProve that $textrank(A) = textrank(A^T)$ using SVDRelationship between eigenvectors and singular vectors of a Hermitian matrix?Image of unit sphere being hyper ellipse proof (SVD)symmetric matrix right and left singular matrix of a symmetric matrixSingular value decomposition inequalityBest singular unit trace matrix approximationProve that rank(A) = rank(A$^dagger$)=rank(AA$^dagger$)=rank(A$^dagger$A) using the SVD decompositionSingular Value Decomposition of a Real Unit Matrix
$begingroup$
The original question image
A $k$-dimensional ellipse, surface and interior, with axes along the standard coordinates is algebraically defined as the set of points $z = (z_1, ldots , z_k)^T$ satisfying $(z_1/alpha_1)^2 + cdots + (z_k/alpha_k)^2 leq 1$.
We can have a $k$-dimensional ellipse embedded inside $mathbbR^n$ even in the case $n > k$ by allowing some of the $z_j$ to be identically zero. Using these definitions, show that the matrix $Sigma = textdiag(sigma_1,...,sigma_min(m,n)) in mathbbR^m times n$, where $sigma_1 geq ldots geq sigma_min (m,n) geq 0$, maps the unit sphere $$, surface and interior, to an ellipse.
Under what conditions is the surface of the unit sphere mapped to the surface of the ellipse? (Suggestion: Consider the cases $m geq n$ and $n > m$ separately. Also, some of the axes of the ellipse may be zero, so it may be convenient to introduce $r leq min(m,n)$ such that $sigma_1 geq ldots geq sigma_r > 0$.)
I think this is like a SVD question and I found the page here that might be useful: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.56.3603&rep=rep1&type=pdf
But I still do not quite understand the exact process of mapping the interior and surface of a unit sphere into an ellipse, maybe SVD is doing what I am asking?
Any help will be greatly appreciated.
matrices svd
asked Apr 7 at 8:02
About_BlanksAbout_Blanks
12
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About_Blanks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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add a comment |
$begingroup$
The original question image
A $k$-dimensional ellipse, surface and interior, with axes along the standard coordinates is algebraically defined as the set of points $z = (z_1, ldots , z_k)^T$ satisfying $(z_1/alpha_1)^2 + cdots + (z_k/alpha_k)^2 leq 1$.
We can have a $k$-dimensional ellipse embedded inside $mathbbR^n$ even in the case $n > k$ by allowing some of the $z_j$ to be identically zero. Using these definitions, show that the matrix $Sigma = textdiag(sigma_1,...,sigma_min(m,n)) in mathbbR^m times n$, where $sigma_1 geq ldots geq sigma_min (m,n) geq 0$, maps the unit sphere $$, surface and interior, to an ellipse.
Under what conditions is the surface of the unit sphere mapped to the surface of the ellipse? (Suggestion: Consider the cases $m geq n$ and $n > m$ separately. Also, some of the axes of the ellipse may be zero, so it may be convenient to introduce $r leq min(m,n)$ such that $sigma_1 geq ldots geq sigma_r > 0$.)
I think this is like a SVD question and I found the page here that might be useful: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.56.3603&rep=rep1&type=pdf
But I still do not quite understand the exact process of mapping the interior and surface of a unit sphere into an ellipse, maybe SVD is doing what I am asking?
Any help will be greatly appreciated.
matrices svd
asked Apr 7 at 8:02
About_BlanksAbout_Blanks
12
New contributor
About_Blanks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
This is a good question, and you've phrased it well. The answer basically has nothing to do with the SVD; it's setting up a more general claim fpor use with the SVD. Here are some questions to ask yourself (and perhaps to answer for yourself in your question, by clicking "edit" at the bottom). Focus on $n = m = 3$ to start. Suppose $T(x) = Sigma x$, where $x in Bbb R^n$. (1) What does it mean (in coordinates) for $x$ to be on or within the unit sphere? (2) Now look at $T(x)$ (in coordinates): what's it mean for $T(x)$ to be on/within the unit sphere? What about on/within the ellipse?
$endgroup$
– John Hughes
Apr 7 at 22:20
$begingroup$
When you've answered those, consider $n = 3, m = 2$. And in both cases, assume that all the $sigma$s are positive -- you can work on the $sigma_i = 0$ cases after you've got the generic case worked out!
$endgroup$
– John Hughes
Apr 7 at 22:21
add a comment |
$begingroup$
The original question image
A $k$-dimensional ellipse, surface and interior, with axes along the standard coordinates is algebraically defined as the set of points $z = (z_1, ldots , z_k)^T$ satisfying $(z_1/alpha_1)^2 + cdots + (z_k/alpha_k)^2 leq 1$.
We can have a $k$-dimensional ellipse embedded inside $mathbbR^n$ even in the case $n > k$ by allowing some of the $z_j$ to be identically zero. Using these definitions, show that the matrix $Sigma = textdiag(sigma_1,...,sigma_min(m,n)) in mathbbR^m times n$, where $sigma_1 geq ldots geq sigma_min (m,n) geq 0$, maps the unit sphere $$, surface and interior, to an ellipse.
Under what conditions is the surface of the unit sphere mapped to the surface of the ellipse? (Suggestion: Consider the cases $m geq n$ and $n > m$ separately. Also, some of the axes of the ellipse may be zero, so it may be convenient to introduce $r leq min(m,n)$ such that $sigma_1 geq ldots geq sigma_r > 0$.)
I think this is like a SVD question and I found the page here that might be useful: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.56.3603&rep=rep1&type=pdf
But I still do not quite understand the exact process of mapping the interior and surface of a unit sphere into an ellipse, maybe SVD is doing what I am asking?
Any help will be greatly appreciated.
matrices svd
asked Apr 7 at 8:02
About_BlanksAbout_Blanks
12
New contributor
About_Blanks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
The original question image
A $k$-dimensional ellipse, surface and interior, with axes along the standard coordinates is algebraically defined as the set of points $z = (z_1, ldots , z_k)^T$ satisfying $(z_1/alpha_1)^2 + cdots + (z_k/alpha_k)^2 leq 1$.
We can have a $k$-dimensional ellipse embedded inside $mathbbR^n$ even in the case $n > k$ by allowing some of the $z_j$ to be identically zero. Using these definitions, show that the matrix $Sigma = textdiag(sigma_1,...,sigma_min(m,n)) in mathbbR^m times n$, where $sigma_1 geq ldots geq sigma_min (m,n) geq 0$, maps the unit sphere $$, surface and interior, to an ellipse.
Under what conditions is the surface of the unit sphere mapped to the surface of the ellipse? (Suggestion: Consider the cases $m geq n$ and $n > m$ separately. Also, some of the axes of the ellipse may be zero, so it may be convenient to introduce $r leq min(m,n)$ such that $sigma_1 geq ldots geq sigma_r > 0$.)
I think this is like a SVD question and I found the page here that might be useful: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.56.3603&rep=rep1&type=pdf
But I still do not quite understand the exact process of mapping the interior and surface of a unit sphere into an ellipse, maybe SVD is doing what I am asking?
Any help will be greatly appreciated.
matrices svd
matrices svd
asked Apr 7 at 8:02
About_BlanksAbout_Blanks
12
New contributor
About_Blanks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Apr 7 at 8:02
About_BlanksAbout_Blanks
12
New contributor
About_Blanks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Apr 7 at 21:35
About_Blanks
asked Apr 7 at 8:02
About_BlanksAbout_Blanks
12
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About_Blanks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Apr 7 at 8:02
About_BlanksAbout_Blanks
12
asked Apr 7 at 8:02
About_BlanksAbout_Blanks
12
12
New contributor
About_Blanks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
About_Blanks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
About_Blanks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
This is a good question, and you've phrased it well. The answer basically has nothing to do with the SVD; it's setting up a more general claim fpor use with the SVD. Here are some questions to ask yourself (and perhaps to answer for yourself in your question, by clicking "edit" at the bottom). Focus on $n = m = 3$ to start. Suppose $T(x) = Sigma x$, where $x in Bbb R^n$. (1) What does it mean (in coordinates) for $x$ to be on or within the unit sphere? (2) Now look at $T(x)$ (in coordinates): what's it mean for $T(x)$ to be on/within the unit sphere? What about on/within the ellipse?
$endgroup$
– John Hughes
Apr 7 at 22:20
$begingroup$
When you've answered those, consider $n = 3, m = 2$. And in both cases, assume that all the $sigma$s are positive -- you can work on the $sigma_i = 0$ cases after you've got the generic case worked out!
$endgroup$
– John Hughes
Apr 7 at 22:21
add a comment |
$begingroup$
This is a good question, and you've phrased it well. The answer basically has nothing to do with the SVD; it's setting up a more general claim fpor use with the SVD. Here are some questions to ask yourself (and perhaps to answer for yourself in your question, by clicking "edit" at the bottom). Focus on $n = m = 3$ to start. Suppose $T(x) = Sigma x$, where $x in Bbb R^n$. (1) What does it mean (in coordinates) for $x$ to be on or within the unit sphere? (2) Now look at $T(x)$ (in coordinates): what's it mean for $T(x)$ to be on/within the unit sphere? What about on/within the ellipse?
$endgroup$
– John Hughes
Apr 7 at 22:20
$begingroup$
When you've answered those, consider $n = 3, m = 2$. And in both cases, assume that all the $sigma$s are positive -- you can work on the $sigma_i = 0$ cases after you've got the generic case worked out!
$endgroup$
– John Hughes
Apr 7 at 22:21
$begingroup$
This is a good question, and you've phrased it well. The answer basically has nothing to do with the SVD; it's setting up a more general claim fpor use with the SVD. Here are some questions to ask yourself (and perhaps to answer for yourself in your question, by clicking "edit" at the bottom). Focus on $n = m = 3$ to start. Suppose $T(x) = Sigma x$, where $x in Bbb R^n$. (1) What does it mean (in coordinates) for $x$ to be on or within the unit sphere? (2) Now look at $T(x)$ (in coordinates): what's it mean for $T(x)$ to be on/within the unit sphere? What about on/within the ellipse?
$endgroup$
– John Hughes
Apr 7 at 22:20
$begingroup$
This is a good question, and you've phrased it well. The answer basically has nothing to do with the SVD; it's setting up a more general claim fpor use with the SVD. Here are some questions to ask yourself (and perhaps to answer for yourself in your question, by clicking "edit" at the bottom). Focus on $n = m = 3$ to start. Suppose $T(x) = Sigma x$, where $x in Bbb R^n$. (1) What does it mean (in coordinates) for $x$ to be on or within the unit sphere? (2) Now look at $T(x)$ (in coordinates): what's it mean for $T(x)$ to be on/within the unit sphere? What about on/within the ellipse?
$endgroup$
– John Hughes
Apr 7 at 22:20
$begingroup$
When you've answered those, consider $n = 3, m = 2$. And in both cases, assume that all the $sigma$s are positive -- you can work on the $sigma_i = 0$ cases after you've got the generic case worked out!
$endgroup$
– John Hughes
Apr 7 at 22:21
$begingroup$
When you've answered those, consider $n = 3, m = 2$. And in both cases, assume that all the $sigma$s are positive -- you can work on the $sigma_i = 0$ cases after you've got the generic case worked out!
$endgroup$
– John Hughes
Apr 7 at 22:21
add a comment |
1 Answer
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$begingroup$
just take an arbitrary point on the surface of the unit sphere, and apply the transformation to it. There will be a condition that results in the transformed point taking on a form of a k-dimensional ellipse.
-Ivo
answered 2 days ago
Ivo PanayotovIvo Panayotov
1
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Ivo Panayotov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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$begingroup$
just take an arbitrary point on the surface of the unit sphere, and apply the transformation to it. There will be a condition that results in the transformed point taking on a form of a k-dimensional ellipse.
-Ivo
answered 2 days ago
Ivo PanayotovIvo Panayotov
1
New contributor
Ivo Panayotov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
just take an arbitrary point on the surface of the unit sphere, and apply the transformation to it. There will be a condition that results in the transformed point taking on a form of a k-dimensional ellipse.
-Ivo
answered 2 days ago
Ivo PanayotovIvo Panayotov
1
New contributor
Ivo Panayotov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
just take an arbitrary point on the surface of the unit sphere, and apply the transformation to it. There will be a condition that results in the transformed point taking on a form of a k-dimensional ellipse.
-Ivo
answered 2 days ago
Ivo PanayotovIvo Panayotov
1
New contributor
Ivo Panayotov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
just take an arbitrary point on the surface of the unit sphere, and apply the transformation to it. There will be a condition that results in the transformed point taking on a form of a k-dimensional ellipse.
-Ivo
answered 2 days ago
Ivo PanayotovIvo Panayotov
1
New contributor
Ivo Panayotov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 days ago
Ivo PanayotovIvo Panayotov
1
New contributor
Ivo Panayotov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 days ago
Ivo PanayotovIvo Panayotov
1
answered 2 days ago
Ivo PanayotovIvo Panayotov
1
1
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$begingroup$
This is a good question, and you've phrased it well. The answer basically has nothing to do with the SVD; it's setting up a more general claim fpor use with the SVD. Here are some questions to ask yourself (and perhaps to answer for yourself in your question, by clicking "edit" at the bottom). Focus on $n = m = 3$ to start. Suppose $T(x) = Sigma x$, where $x in Bbb R^n$. (1) What does it mean (in coordinates) for $x$ to be on or within the unit sphere? (2) Now look at $T(x)$ (in coordinates): what's it mean for $T(x)$ to be on/within the unit sphere? What about on/within the ellipse?
$endgroup$
– John Hughes
Apr 7 at 22:20
$begingroup$
When you've answered those, consider $n = 3, m = 2$. And in both cases, assume that all the $sigma$s are positive -- you can work on the $sigma_i = 0$ cases after you've got the generic case worked out!
$endgroup$
– John Hughes
Apr 7 at 22:21