Mixing
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Matrices and Manifolds
$begingroup$
I have a huge problem solving this question. Does anybody have literature on this or maybe a solution?
Let
$
M:={xin mathbb{R}^n : x^{⊤} A x = r} subset mathbb{R}^n
$
for $r>0$ and $A$ being a symmetric, positive semidefinite Matrix $Ain mathbb{R}^{n times n}$
Is $M$ a $C^l$-Submanifold of $mathbb{R}^n$. If so, of which Dimension?
analysis manifolds submanifold
asked Jan 13 at 21:53
KingDingelingKingDingeling
1517
$endgroup$
add a comment |
$begingroup$
I have a huge problem solving this question. Does anybody have literature on this or maybe a solution?
Let
$
M:={xin mathbb{R}^n : x^{⊤} A x = r} subset mathbb{R}^n
$
for $r>0$ and $A$ being a symmetric, positive semidefinite Matrix $Ain mathbb{R}^{n times n}$
Is $M$ a $C^l$-Submanifold of $mathbb{R}^n$. If so, of which Dimension?
analysis manifolds submanifold
asked Jan 13 at 21:53
KingDingelingKingDingeling
1517
$endgroup$
add a comment |
$begingroup$
I have a huge problem solving this question. Does anybody have literature on this or maybe a solution?
Let
$
M:={xin mathbb{R}^n : x^{⊤} A x = r} subset mathbb{R}^n
$
for $r>0$ and $A$ being a symmetric, positive semidefinite Matrix $Ain mathbb{R}^{n times n}$
Is $M$ a $C^l$-Submanifold of $mathbb{R}^n$. If so, of which Dimension?
analysis manifolds submanifold
asked Jan 13 at 21:53
KingDingelingKingDingeling
1517
$endgroup$
I have a huge problem solving this question. Does anybody have literature on this or maybe a solution?
Let
$
M:={xin mathbb{R}^n : x^{⊤} A x = r} subset mathbb{R}^n
$
for $r>0$ and $A$ being a symmetric, positive semidefinite Matrix $Ain mathbb{R}^{n times n}$
Is $M$ a $C^l$-Submanifold of $mathbb{R}^n$. If so, of which Dimension?
analysis manifolds submanifold
analysis manifolds submanifold
asked Jan 13 at 21:53
KingDingelingKingDingeling
1517
asked Jan 13 at 21:53
KingDingelingKingDingeling
1517
edited Jan 13 at 21:54
KingDingeling
asked Jan 13 at 21:53
KingDingelingKingDingeling
1517
asked Jan 13 at 21:53
KingDingelingKingDingeling
1517
asked Jan 13 at 21:53
KingDingelingKingDingeling
1517
1517
add a comment |
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
There exists a symmetric positive semidefinite matrix $B$ such that $B^2=A$. Let $K$ denote the set of vectors $x$ such that $Bx=0$. Let $L$ be its orthogonal: $B$ induces an isomorphism on $L$.
Then the linear isomorphism $mathbb{R}^n rightarrow K times L$ maps $M$ to $K times B^{-1}mathcal{S}_L(r)$. Thus $M$ is a submanifold of dimension $n-1$.
answered Jan 13 at 22:00
MindlackMindlack
4,450210
$endgroup$
1
$begingroup$
Thanks a lot. Is there maybe literature about this? Has this Set a name or something?
$endgroup$
– KingDingeling
Jan 13 at 22:05
1
$begingroup$
Not to my knowledge, but I am afraid I do not know much about the literature.
$endgroup$
– Mindlack
Jan 13 at 22:53
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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$begingroup$
There exists a symmetric positive semidefinite matrix $B$ such that $B^2=A$. Let $K$ denote the set of vectors $x$ such that $Bx=0$. Let $L$ be its orthogonal: $B$ induces an isomorphism on $L$.
Then the linear isomorphism $mathbb{R}^n rightarrow K times L$ maps $M$ to $K times B^{-1}mathcal{S}_L(r)$. Thus $M$ is a submanifold of dimension $n-1$.
answered Jan 13 at 22:00
MindlackMindlack
4,450210
$endgroup$
1
$begingroup$
Thanks a lot. Is there maybe literature about this? Has this Set a name or something?
$endgroup$
– KingDingeling
Jan 13 at 22:05
1
$begingroup$
Not to my knowledge, but I am afraid I do not know much about the literature.
$endgroup$
– Mindlack
Jan 13 at 22:53
add a comment |
$begingroup$
There exists a symmetric positive semidefinite matrix $B$ such that $B^2=A$. Let $K$ denote the set of vectors $x$ such that $Bx=0$. Let $L$ be its orthogonal: $B$ induces an isomorphism on $L$.
Then the linear isomorphism $mathbb{R}^n rightarrow K times L$ maps $M$ to $K times B^{-1}mathcal{S}_L(r)$. Thus $M$ is a submanifold of dimension $n-1$.
answered Jan 13 at 22:00
MindlackMindlack
4,450210
$endgroup$
1
$begingroup$
Thanks a lot. Is there maybe literature about this? Has this Set a name or something?
$endgroup$
– KingDingeling
Jan 13 at 22:05
1
$begingroup$
Not to my knowledge, but I am afraid I do not know much about the literature.
$endgroup$
– Mindlack
Jan 13 at 22:53
add a comment |
$begingroup$
There exists a symmetric positive semidefinite matrix $B$ such that $B^2=A$. Let $K$ denote the set of vectors $x$ such that $Bx=0$. Let $L$ be its orthogonal: $B$ induces an isomorphism on $L$.
Then the linear isomorphism $mathbb{R}^n rightarrow K times L$ maps $M$ to $K times B^{-1}mathcal{S}_L(r)$. Thus $M$ is a submanifold of dimension $n-1$.
answered Jan 13 at 22:00
MindlackMindlack
4,450210
$endgroup$
There exists a symmetric positive semidefinite matrix $B$ such that $B^2=A$. Let $K$ denote the set of vectors $x$ such that $Bx=0$. Let $L$ be its orthogonal: $B$ induces an isomorphism on $L$.
Then the linear isomorphism $mathbb{R}^n rightarrow K times L$ maps $M$ to $K times B^{-1}mathcal{S}_L(r)$. Thus $M$ is a submanifold of dimension $n-1$.
answered Jan 13 at 22:00
MindlackMindlack
4,450210
answered Jan 13 at 22:00
MindlackMindlack
4,450210
answered Jan 13 at 22:00
MindlackMindlack
4,450210
answered Jan 13 at 22:00
MindlackMindlack
4,450210
4,450210
1
$begingroup$
Thanks a lot. Is there maybe literature about this? Has this Set a name or something?
$endgroup$
– KingDingeling
Jan 13 at 22:05
1
$begingroup$
Not to my knowledge, but I am afraid I do not know much about the literature.
$endgroup$
– Mindlack
Jan 13 at 22:53
add a comment |
1
$begingroup$
Thanks a lot. Is there maybe literature about this? Has this Set a name or something?
$endgroup$
– KingDingeling
Jan 13 at 22:05
1
$begingroup$
Not to my knowledge, but I am afraid I do not know much about the literature.
$endgroup$
– Mindlack
Jan 13 at 22:53
1
1
$begingroup$
Thanks a lot. Is there maybe literature about this? Has this Set a name or something?
$endgroup$
– KingDingeling
Jan 13 at 22:05
$begingroup$
Thanks a lot. Is there maybe literature about this? Has this Set a name or something?
$endgroup$
– KingDingeling
Jan 13 at 22:05
1
1
$begingroup$
Not to my knowledge, but I am afraid I do not know much about the literature.
$endgroup$
– Mindlack
Jan 13 at 22:53
$begingroup$
Not to my knowledge, but I am afraid I do not know much about the literature.
$endgroup$
– Mindlack
Jan 13 at 22:53
add a comment |
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