Proof: A tangent space of the manifold of SPD matrices is the set of symmetric matrices
$begingroup$
The set of SPD matrices, $mathbb{P}_n := {X in mathbb{R}^{n times n} | X=X^T, X succ 0 } $, forms a differentiable manifold.
Claim: The tangent space at a point, $A, T_Amathcal{P}_n$ is the space of symmetric matrices.
Info: I have seen this mentioned in a paper (https://www.ncbi.nlm.nih.gov/pubmed/27845666), but (with my limited understanding of differential geometry) can't see how this occurs (although it may actualy be obvious to someone with a more rigorous maths education). I have also checked out the book they referenced when making this claim (http://www.cmat.edu.uy/~lessa/tesis/Positive%20Definite%20Matrices.pdf) in order to understand it. Sadly the section that I think they are referencing (Start of chapter 6) is far too formal for me. For completeness I will write out what is written there, and if it is the proof I am looking for then an explanation of some of the jumps in the steps would answer my question.
1) The space $mathbb{M}_n$ is a Hilbert space with inner product $ langle A,B rangle = tr A^*B $ and the associated norm $vert vert A vert vert _2 = (tr A^*A)^{frac{1}{2}}$
2) The set of Hermitian matrices constitutes a real vector space $mathbb{H}_n$ in $mathbb{M}_n$.
3) The subset $mathbb{P}_n$ consisting of strictly positive matrices is an open subset in $mathbb{H}_n$. Hence it is a differentiable manifold.
4) The tangent space to $mathbb{P}_n$ at any of its points A is the space $T_Amathbb{P}_n = {A} times mathbb{H}_n$
If this is indeed the proof that I seek then these are the steps that I don't understand:
3) I know that open subsets of manifolds are manifolds, and I am also presuming that "stricly positive" means positive definite. But how do I know that this manifold is differentiable, ie has differentiable transistion maps?
4) This is my main confusion - $mathbb{H}_n$ is the set of symmetric matrices, but why is the tangent space at any point defined like this? I understand the tangent space to be the collection of velocity vectors through that point. Why can I represent the gradients of all the geodesics that pass through a point like this?
EDIT - UPDATE:
Having been browsing the stack I also came across tangent space clarification where someone is asking why tangent spaces aren't just defined in the same way (ish) as they were in 4) - does anyone know where this definition/way of writing has come from?
differential-geometry proof-writing manifolds proof-explanation positive-definite
asked Dec 13 '18 at 10:27
bidbybidby
976
$endgroup$
add a comment |
$begingroup$
The set of SPD matrices, $mathbb{P}_n := {X in mathbb{R}^{n times n} | X=X^T, X succ 0 } $, forms a differentiable manifold.
Claim: The tangent space at a point, $A, T_Amathcal{P}_n$ is the space of symmetric matrices.
Info: I have seen this mentioned in a paper (https://www.ncbi.nlm.nih.gov/pubmed/27845666), but (with my limited understanding of differential geometry) can't see how this occurs (although it may actualy be obvious to someone with a more rigorous maths education). I have also checked out the book they referenced when making this claim (http://www.cmat.edu.uy/~lessa/tesis/Positive%20Definite%20Matrices.pdf) in order to understand it. Sadly the section that I think they are referencing (Start of chapter 6) is far too formal for me. For completeness I will write out what is written there, and if it is the proof I am looking for then an explanation of some of the jumps in the steps would answer my question.
1) The space $mathbb{M}_n$ is a Hilbert space with inner product $ langle A,B rangle = tr A^*B $ and the associated norm $vert vert A vert vert _2 = (tr A^*A)^{frac{1}{2}}$
2) The set of Hermitian matrices constitutes a real vector space $mathbb{H}_n$ in $mathbb{M}_n$.
3) The subset $mathbb{P}_n$ consisting of strictly positive matrices is an open subset in $mathbb{H}_n$. Hence it is a differentiable manifold.
4) The tangent space to $mathbb{P}_n$ at any of its points A is the space $T_Amathbb{P}_n = {A} times mathbb{H}_n$
If this is indeed the proof that I seek then these are the steps that I don't understand:
3) I know that open subsets of manifolds are manifolds, and I am also presuming that "stricly positive" means positive definite. But how do I know that this manifold is differentiable, ie has differentiable transistion maps?
4) This is my main confusion - $mathbb{H}_n$ is the set of symmetric matrices, but why is the tangent space at any point defined like this? I understand the tangent space to be the collection of velocity vectors through that point. Why can I represent the gradients of all the geodesics that pass through a point like this?
EDIT - UPDATE:
Having been browsing the stack I also came across tangent space clarification where someone is asking why tangent spaces aren't just defined in the same way (ish) as they were in 4) - does anyone know where this definition/way of writing has come from?
differential-geometry proof-writing manifolds proof-explanation positive-definite
asked Dec 13 '18 at 10:27
bidbybidby
976
$endgroup$
add a comment |
$begingroup$
The set of SPD matrices, $mathbb{P}_n := {X in mathbb{R}^{n times n} | X=X^T, X succ 0 } $, forms a differentiable manifold.
Claim: The tangent space at a point, $A, T_Amathcal{P}_n$ is the space of symmetric matrices.
Info: I have seen this mentioned in a paper (https://www.ncbi.nlm.nih.gov/pubmed/27845666), but (with my limited understanding of differential geometry) can't see how this occurs (although it may actualy be obvious to someone with a more rigorous maths education). I have also checked out the book they referenced when making this claim (http://www.cmat.edu.uy/~lessa/tesis/Positive%20Definite%20Matrices.pdf) in order to understand it. Sadly the section that I think they are referencing (Start of chapter 6) is far too formal for me. For completeness I will write out what is written there, and if it is the proof I am looking for then an explanation of some of the jumps in the steps would answer my question.
1) The space $mathbb{M}_n$ is a Hilbert space with inner product $ langle A,B rangle = tr A^*B $ and the associated norm $vert vert A vert vert _2 = (tr A^*A)^{frac{1}{2}}$
2) The set of Hermitian matrices constitutes a real vector space $mathbb{H}_n$ in $mathbb{M}_n$.
3) The subset $mathbb{P}_n$ consisting of strictly positive matrices is an open subset in $mathbb{H}_n$. Hence it is a differentiable manifold.
4) The tangent space to $mathbb{P}_n$ at any of its points A is the space $T_Amathbb{P}_n = {A} times mathbb{H}_n$
If this is indeed the proof that I seek then these are the steps that I don't understand:
3) I know that open subsets of manifolds are manifolds, and I am also presuming that "stricly positive" means positive definite. But how do I know that this manifold is differentiable, ie has differentiable transistion maps?
4) This is my main confusion - $mathbb{H}_n$ is the set of symmetric matrices, but why is the tangent space at any point defined like this? I understand the tangent space to be the collection of velocity vectors through that point. Why can I represent the gradients of all the geodesics that pass through a point like this?
EDIT - UPDATE:
Having been browsing the stack I also came across tangent space clarification where someone is asking why tangent spaces aren't just defined in the same way (ish) as they were in 4) - does anyone know where this definition/way of writing has come from?
differential-geometry proof-writing manifolds proof-explanation positive-definite
asked Dec 13 '18 at 10:27
bidbybidby
976
$endgroup$
The set of SPD matrices, $mathbb{P}_n := {X in mathbb{R}^{n times n} | X=X^T, X succ 0 } $, forms a differentiable manifold.
Claim: The tangent space at a point, $A, T_Amathcal{P}_n$ is the space of symmetric matrices.
Info: I have seen this mentioned in a paper (https://www.ncbi.nlm.nih.gov/pubmed/27845666), but (with my limited understanding of differential geometry) can't see how this occurs (although it may actualy be obvious to someone with a more rigorous maths education). I have also checked out the book they referenced when making this claim (http://www.cmat.edu.uy/~lessa/tesis/Positive%20Definite%20Matrices.pdf) in order to understand it. Sadly the section that I think they are referencing (Start of chapter 6) is far too formal for me. For completeness I will write out what is written there, and if it is the proof I am looking for then an explanation of some of the jumps in the steps would answer my question.
1) The space $mathbb{M}_n$ is a Hilbert space with inner product $ langle A,B rangle = tr A^*B $ and the associated norm $vert vert A vert vert _2 = (tr A^*A)^{frac{1}{2}}$
2) The set of Hermitian matrices constitutes a real vector space $mathbb{H}_n$ in $mathbb{M}_n$.
3) The subset $mathbb{P}_n$ consisting of strictly positive matrices is an open subset in $mathbb{H}_n$. Hence it is a differentiable manifold.
4) The tangent space to $mathbb{P}_n$ at any of its points A is the space $T_Amathbb{P}_n = {A} times mathbb{H}_n$
If this is indeed the proof that I seek then these are the steps that I don't understand:
3) I know that open subsets of manifolds are manifolds, and I am also presuming that "stricly positive" means positive definite. But how do I know that this manifold is differentiable, ie has differentiable transistion maps?
4) This is my main confusion - $mathbb{H}_n$ is the set of symmetric matrices, but why is the tangent space at any point defined like this? I understand the tangent space to be the collection of velocity vectors through that point. Why can I represent the gradients of all the geodesics that pass through a point like this?
EDIT - UPDATE:
Having been browsing the stack I also came across tangent space clarification where someone is asking why tangent spaces aren't just defined in the same way (ish) as they were in 4) - does anyone know where this definition/way of writing has come from?
differential-geometry proof-writing manifolds proof-explanation positive-definite
differential-geometry proof-writing manifolds proof-explanation positive-definite
asked Dec 13 '18 at 10:27
bidbybidby
976
asked Dec 13 '18 at 10:27
bidbybidby
976
edited Dec 13 '18 at 14:24
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asked Dec 13 '18 at 10:27
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asked Dec 13 '18 at 10:27
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asked Dec 13 '18 at 10:27
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