Tagged Questions

2
votes
3answers
45 views

symmetric positive definite matrices

Why must a symmetric positive definite matrix must be invertible? I'm reading a proof of the Levi-Civita theorem in differential geometry but the author states this without proof and I haven't been ...
1
vote
1answer
38 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 ...
3
votes
2answers
71 views

How to quantify the distance between matrices with an irrelevant rotation factor?

Suppose you have two invertible matrices $A$, $B$ in $\mathbb{R}^{n\times n}$, that is, $A,B\in GL(n)$. You want to define a distance between them that ignores arbitrary rotational factors, so ...
3
votes
3answers
135 views

Fit a quadratic form given covariant derivatives on the sphere?

I am trying to solve for a particular vector given covariant first and second derivative for a function on a sphere. If you have a quadratic form restricted to the sphere: $f(x) = ...
5
votes
1answer
103 views

Invariant Inner Product on Lie Algebra

Let $G$ be a Lie group, $\frak{g}$ its Lie algebra. Suppose $\mathcal{D}$ a representation of $G$ on $V$, $d$ the associated Lie algebra representation. Suppose $V$ is endowed with an inner product. ...
1
vote
1answer
67 views

Explanation of easy statement regarding derivative and Jacobian needed

Let $\Phi:S \to T$ be a map between surfaces in $\mathbb{R}^n$. What precisely does this mean: Let $\text{det}(\mathbf{D}_S \Phi(.))$ denote the Jacobian determinant of the matrix representation ...
0
votes
1answer
123 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 ...
3
votes
2answers
392 views

How to find the tangent space to a matrix space

I have a hard time approaching these types of problems. In an article it had claimed that the tangent space to all symmetric matrices with the same signature as $M$ at a matrix $M$ is the set of all ...
1
vote
1answer
54 views

to show $g$ attains maxima and minima

Let $A$ be a symmetric $n\times n$ real matrix and define $G:\mathbb{R}^n\rightarrow \mathbb{R}$ by $G(t)=\langle At,t\rangle$; let $g:S^{n-1}\rightarrow \mathbb{R}$ be the restriction of $G$ to the ...
1
vote
1answer
316 views

The square root of positive definite matrix

Let $M$ be the manifold of real positive definite $n \times n$ matrices, define a mapping $i:A \to \sqrt A$ (where $A\in M$ and $\sqrt A$ means the unique positive definite square root of $A$). Please ...
2
votes
2answers
267 views

Proof: Tangent space of the general linear group is the set of all squared matrices

Let us assume we have the following definition of a tangent space: Definition of smooth path Let $X\subset\mathbb{R}^n$. Let $I$ be a real interval. \begin{equation} P \text{ is a smooth path in } ...
4
votes
1answer
235 views

Lebesgue measure on normal matrices

Consider the space of $n\times n$ complex matrices, and equip it with its Lebesgue measure $dX$, seen as a $2n^2$-dimensional real vector space [edit: or better, a complex vector space (see the answer ...
0
votes
0answers
179 views

Rotation matrix from Frenet frame

I don't know, how I can generate a rotation matrix from a Frenet view. The result must be a $3 \times 3$-matrix. I have all orientation vectors, $\verb|upVector|$, $\verb|viewVector|$ and the ...