1
vote
1answer
38 views

Rotation Matrix to Quaternion(proper Orientation)

Given Data in the figure In this figure we have a unit vectors $ x,y,z$ as axis. Axis of rotation is $b$ and angle of rotation is $\phi$. $\phi$ is unknown and $b$ is given as $b= \frac{1}{2 ...
2
votes
0answers
80 views

Quaternion conversion

We have a normalized orthogonal co-ordinate frame travelling through the curve as in figure 1 below, from one end to other. Let us call starting end as A and ending end as B. What we know is initial ...
0
votes
0answers
14 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
0
votes
0answers
30 views

Matrices with functions as entries

I am interested is studying matrices which have functional entries. Specifically I am looking at quadratic forms of the type $x^T Q(x) x$ where $Q(x)$ is a matrix whose entries are functions of $x$. I ...
1
vote
1answer
48 views

Is this a tangent bundle and what is the meaning of this exercise

I intend to solve the following exercise but I would like to have some help with understanding the ''big picture'': Exercise. Describe a natural 1 to 1 correspondence between elements of $SO(3)$ ...
1
vote
1answer
48 views

Left-invariant Riemannian metric on $SO(3)$

Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one ...
3
votes
1answer
61 views

tangent/normal space to set of symmetric isospectral matrices

Let $\Lambda = \{\lambda_1, \ldots, \lambda_n\}$ be a set of $n$ distinct real numbers. $M_n(\mathbb{R})$ denotes the set of all $n \times n$ real matrices, and for $B\in M_n(\mathbb{R})$, $B^T$ ...
0
votes
2answers
48 views

Need help finding Jacobian matrix of diffeomorphism of spheres

Let $S_a \subset \mathbb{R}^{n+1}$ and $S_b \subset \mathbb{R}^{n+1}$ be two spheres of radius $a$ and $b$ respectively. So $S_a$ are $n$-dimensional. Let $F:S_a \to S_b$ be the diffeomorphism $F(s) ...
21
votes
1answer
748 views

How to find eigenvalues and eigenvectors of this matrix

Can you help to find eigenvalues and eigenvectors of the following matrix? Here is the matrix: $$C = \small \begin{pmatrix} -\sin(\theta_{2} - \theta_{M}) & \sin(\theta_{1} - \theta_{M}) & 0 ...
2
votes
2answers
53 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 ...
2
votes
1answer
44 views

Parameter Transformation with the Jacobian

If $\phi:U\rightarrow V$ and $\tilde{\phi}:\tilde{U}\rightarrow\tilde{V}$ are parametrizations of a regular surface $S$ with $V\cap\tilde{V}≠0$ and $V, \tilde{V}\subset S$. Let $E,F,G$ and ...
0
votes
0answers
17 views

Scalar Product of 2 Matrices

Scalar Product of 2 Matrices If we have for example; $f:J\times\mathbb R\rightarrow\mathbb R^3 $,$\qquad$$(x,y)\mapsto(g(x), h(x)cos(y),h(x)sin(y))$ and $\alpha:I\rightarrow J\times\mathbb ...
0
votes
0answers
42 views

Integration by substitution and transformation

Let $B_r \subset \mathbb{R}^{n-1}$ be a ball of radius $r$ centred at $0$. Let $h \in C^{0,1}(\overline{B_r})$. Consider $$I=\int_{B_r}u\big(\phi(y_1, ..., y_{n-1}, h(y_1, ..., ...
2
votes
1answer
109 views

Differentiating the determinant of the Jacobian of a diffeomorphism (don't understand a proof)

For each $t$, let $A_t:\Omega_0 \to \Omega_t$ be a bi-Lipschitz map between open sets in $\mathbb{R}^n$. The map is also invertible. It satisfies $$\frac{d}{dt}A_t(y) = w(A_t(y),t)$$ where $w$ is a ...
3
votes
3answers
211 views

What is the dimension of this Grassmannian?

Why is $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension? Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all ...
1
vote
1answer
103 views

derivative of a positive definite matrix

Suppose that $A$ is a positive definite symmetric matrix, specifically a Riemannian metric. Can we say anything about the sign of $tr(A^{-1}\partial_i A)$?
1
vote
1answer
93 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
2answers
77 views

behavior of scalar product defined by trace under commutator

Define $$\langle X,Y \rangle := \operatorname{tr}XY^t,$$ where $X,Y$ are square matrices with real entries and $t$ denotes transpose. I have some troubles in proving that $$ \langle [X,Y],Z \rangle = ...
8
votes
1answer
212 views

Maurer-Cartan 1-form

Can anyone help me with the following? Let $\rho$ be the right-invariant Maurer-Cartan 1-form $$\rho = dg\ g^{-1}$$ I want to show that the MC equation $$d\rho - \rho \wedge\rho = 0$$ holds. So ...
2
votes
3answers
124 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
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 ...
3
votes
2answers
404 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
151 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
284 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
104 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 ...
1
vote
1answer
194 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
949 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
67 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 ...
3
votes
1answer
481 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
663 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
355 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 ...