2
votes
1answer
27 views

Givens rotation and retraction mapping

Here's part of the texts in the book Optimization Algorithms on Matrix Manifolds, when talking about the retraction on a matrix sub-manifold. A retraction is a mapping that maps a tangent vector in a ...
2
votes
1answer
40 views

Proving a set of $2\times 3$ matrices is a manifold?

The way I have always been told to check if something is a manifold (I haven't had a whole lot of experience with them), is to check if the derivative of the function representing the loci of the ...
2
votes
2answers
45 views

Hessian of Morse function on $S^{n}$ mistake

I am trying to get that $f(x_{0},...,x_{n+1})=x_{n+1}$ has $index_{(0,...,0,1)}=n$ Can you find my mistake or post a partial solution? My attempt I evaluate df using inverse of stereographic ...
1
vote
1answer
89 views

Set of matrices differentiable manifold? [closed]

Let $X$ be a set of matrices $2\times 3$, that for all $A$ from $X$ rank $A=1$. Is $X$ a manifold? If not find a maximum subset in $X$, which is a manifold and its dimension.
1
vote
1answer
32 views

Kernel of matrix with identity as submatrix

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^k$ be a $C^\infty$ map and let $X=\text{graph}f$, i.e. $$X=\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^k\mid y=f(x)\}.$$ What is the tangent space to $X$ at ...
3
votes
1answer
42 views

Derivative of function between sets of matrices

Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Let $\phi:M_{k,n}\rightarrow S_k$ be the map ...
0
votes
0answers
20 views

Prove that if $A$ and $B$ are two $k$-frames that there is an invertible $n\times n$ matrix $M$ such that $M\cdot A=B$.

Ok, so a $k$-frame (in case this isn't common terminology) is an $n\times k$ matrix with rank $k$. I realize that an invertible $n\times n$ matrix, when multiplied on the left by a $k$-frame, will ...
1
vote
1answer
89 views

Jacobian of matrix product in SU(n)

I need to compute the determinant of the jacobian matrix of the function $f: SO(n)\times SO(n) \rightarrow SO(n)\times SO(n)$ given by $f(P,V) = (PV, P^TV)$. I've found the jacobian if we extend $f$ ...
1
vote
1answer
41 views

Is $VV^T + D$ a submanifold?

If the positive definite matrix P forms a manifold, is that the subset that {P: P = V V^T + D} where V is a low rank matrix and D is a positive definite matrix a sub-manifold? This idea is ...
1
vote
0answers
75 views

transformation of symplectic structure by a matrix

Suppose that in canonical symplectic basis $e_1,e_2,f_1,f_2$ we have $$\Omega=pf_1^*\wedge f_2^*+qe_1^*\wedge e_2^*+r(e_1^*\wedge f_2^*+e_2^*\wedge f_1^*)+s(e_1^*\wedge f_1^*-e_2^*\wedge f_2^*)$$ Let ...
2
votes
0answers
124 views

Questions about algebraic manifold on matrices

The following snapshot comes from the paper Latent Variable Graphical Model Selection Via Convex Optimization: I know little about algebraic geometry so I have several basic questions: How is the ...
1
vote
1answer
182 views

Dimension of the space of matrices with constant determinant.

I'm looking for the dimension of the space of $n\times n$ real matrices $A$ such that $\det(A)=c$. I apply 2 different approaches and I get different answers. which one is correct? 1) So we ...
1
vote
1answer
303 views

Symmetric, upper triangular, diagonal and null-trace matrix spaces: are they manifolds?

I have to prove that to each of following classes of matrices can be given a manifold structure: symmetric (denoted with $\mathcal{S}$) upper triangular diagonal null trace. I am interested in ...
2
votes
1answer
96 views

Real square matrices space as a manifold

Let $\mathrm{M}(n,\mathbb{R})$ be the space of $n\times n$ matrices over $\mathbb{R}$. Consider the function $m \in \mathrm{M}(n,\mathbb{R}) \mapsto (a_{11},\dots,a_{1n},a_{21}\dots,a_{nn}) \in ...