All Questions
Tagged with mg.metric-geometry linear-algebra
97 questions
3
votes
1
answer
375
views
Connections between a polytope's symmetry group and the existence of periodic orbits
Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with ...
- 269
3
votes
1
answer
1k
views
Find a line such that sum of perpendicular distances of points to the line is minimized
Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
user49129
3
votes
0
answers
81
views
Iterated crossproducts
Let $S_0$ be a set of vectors in $\mathbb{R}^n$. Now iteratively define $S_{j+1}$ by taking all possible crossproducts of vectors in $S_j$ and normalizing such that the vector of maximum norm has unit ...
- 101
3
votes
0
answers
526
views
Cavalieri's principle and inversion of the Vandermonde matrix
There are many examples on the Web of the use of Cavalieri's principle in determining areas and volumes of 2-D and 3-D geometrical figures. The Wikipedia link uses the principle as both a proof and ...
- 10.5k
3
votes
0
answers
170
views
Is there such a matrix in $SO(n)$?
Given two $n$ dimensional positive definite matrices $A', B'$, is there a matrix $O \in SO(n)$ such that $A=O A', B=O B'$ and
$$
\frac{A_{ij}}{\sqrt{A_{ii}A_{jj}}} = \frac{B_{ij}}{\sqrt{B_{ii}B_{jj}}},...
- 131
3
votes
0
answers
75
views
Are there a group of mappings from (n-1)-dim space to an (n-1)-sphere guaranteeing the orthogonality of images?
Hello, everyone.
As we know that in an $n$-dimensional Euclidean space $\mathbb{R}^n$, there exists a continuous bijective mapping from a subset $V^{n-1}\subseteq\mathbb{R}^{n-1}$ to a unit $(n-1)$-...
- 607
2
votes
3
answers
355
views
Geometric means of matrices beyond the positive definite cone
Recently a lot of work has been done on geometric means of positive definite matrices (see here and here for example). Has anyone extended this concept to larger sets of matrices (copositive, for ...
- 6,962
2
votes
1
answer
312
views
Question on a vector inequality
Is it true that
$$
\min\left( \begin{aligned}
&\|\mathbf{u}\| + \|\mathbf{v}\| - \|\mathbf{u} + \mathbf{v}\|, \\
&\|\mathbf{u}\| + \|\mathbf{w}\| - \|\mathbf{u} + \mathbf{w}\|, \\
&\|\...
- 171
2
votes
2
answers
164
views
Looking for a simple proof that the generalized disc is bounded
So let us define the generalized disc of degree $n$ as
$$
\mathbb{D}_n:=\{w\in M_{n\times n}(\mathbb{C}):w=w^t, I_n-w\overline{w}>0\}.
$$
For a Hermitian matrix $A$, the notation $A>0$ means ...
- 7,609
2
votes
2
answers
248
views
On an angle distribution of a random linear subspace of a given dimension
$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
- 128k
2
votes
1
answer
384
views
Feasible space of SDP
Typically the non-empty feasible space of a SDP has some curved boundary which is why the feasible space has infinitely many extreme points. Is it ever possible to have a SDP whose non-empty feasible ...
- 505
2
votes
2
answers
273
views
Johnson-Lindenstrauss Lemma on $S^{d-1}$
Consider the Johnson-Lindenstrauss lemma in the case where we can assume the $n$ input points $x_i$ in $\mathbb{R}^d$ are actually located on the sphere
$$S^{d-1}(r):=\{u=(u_1,\ldots,u_{d}): u_1^2+\...
- 10.4k
2
votes
1
answer
237
views
Geometric interpretation of trace of a linear operator
This question is really an addendum to Geometric interpretation of trace
There is a nice account of the trace in Chris Doran's thesis here: http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/...
- 21
2
votes
0
answers
38
views
Constructing an $n$-simplex at the border of a $n$-ball by orthogonal hyperplanes
I want to construct an $n$-simplex the following way:
Choose $n$ vectors in the boundary of an $n$ dimensional ball, which are forming an $(n-1)$-simplex together.
Place the orthogonal affine $n-1$-...
2
votes
0
answers
233
views
Do you know this formula for the scalar product in barycentric coordinates?
I've found a formula for a scalar product in barycentric coordinates which I think is pretty cool. I hope that it's new. Is it?
Suppose that you have points $x_1,\dots,x_n$ sitting in general position ...
- 1,048
2
votes
0
answers
46
views
signatures of quasi-gram matrices
Suppose I have a finite subset $\mathcal{M}$ of a Banach space $B$ $\mathcal{M}=p_1, \dots, p_n,$ and I create the following ``Gram'' matrix $G_{\mathcal{M}}:$
$$g_{ij} = \frac{\|p_i\|^2 + \|p_j\|^2 ...
- 96.4k
2
votes
0
answers
51
views
Find Line through the origin which minimizes the squared perpendicular distances to given points [closed]
Given a set of Points $p_i\in \Re^d$, I'm looking for the vector $x\in\Re^d$ with $\| x \|=1$ along a line so that minimizes $$\sum_{i}(\|p_i\|^2 - \langle p_i, x\rangle^2)$$
According to the first ...
- 21
2
votes
0
answers
48
views
Minimization of the volume of the image of space-filling convex polytopes under similarities
Suppose $A:\mathbb{R}^n \to \mathbb{R}^n$ is a similarity, given by $A(x) = \lambda Ox$, where $\lambda > 1$ and $O$ is an orthogonal matrix (i.e., $A$ is a particular loxodromic repelling ...
- 21
2
votes
0
answers
92
views
Estimating the size of a subset of $\mathbb{R}^N$
This concrete geometric question has arisen out of the problem of counting arithmetic functions with a particular property. The details of the relationship between the counting procedure and this ...
- 2,480
2
votes
0
answers
529
views
Good covering of a sphere
Consider a sphere $S_r(0)$ with center at zero and radius $r$ in the Hamming space $\{0,1\}^n$.
We will be interested in covering this sphere with balls of radius $\rho < r$.
We know that there ...
- 2,576
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
1
vote
1
answer
132
views
Can I find $n$ points on the boundary of an $n$-dimensional ball with certain properties?
My problem is the following: I want to construct $n$ rays all starting at a point $v$ that is not in the $n$-dimensional ball around $0$ such that the following is true:
The $n$-dimensional ball is a ...
1
vote
2
answers
1k
views
Möbius transformation by 3 points in the Minkowski model
Goal
I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.
What I have tried
I know that a projective ...
- 534
1
vote
1
answer
81
views
Generate an ordered set of mostly orthogonal vectors $\{x_i\}$ where $x_i \cdot x_j =0$ iff $\lvert i-j\rvert >m$
I am wondering if there is a way to formulate or generate a matrix $X \in R^{n\times n}$ whose column vectors $\{x_1,\dotsc,x_i,\dotsc,x_n\}$ are such that $x_i$ and $x_j$ are orthogonal iff $\lvert i-...
- 433
1
vote
1
answer
137
views
Relationships among lattices U14, C2xG23, A15+ and their Delaunay polytopes
Do you have any references explaining the relationships among the lattices U14, C2xG23 aka Q14, and A15+?
Do you have any references explaining the relationships among these lattices and the 7D ...
- 53
1
vote
1
answer
184
views
Average distance between points of lower dimensional simplices in $\mathbb R^n$
Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...
- 6,155
1
vote
1
answer
428
views
Covering number in the space of symmetric matrices
Let $S_n(\mathbb{R})$ be the set of symmetric matrices of size $n \times n$. Note $\|\Theta\|_{0}$ the number of nonzero elements of a matrix $\Theta$ and $\|\cdot\|_F$ the Froebenius norm. Consider ...
- 407
1
vote
1
answer
319
views
Are there any applications of linear algebra over the complex numbers, where the role of complex conjugation is replaced with the trivial involution?
The complex inner product $\langle u, v \rangle$ is a special case of a sesquilinear form over a field. Its definition is $\langle u, v \rangle = \sum_{i} u_i \overline{v_i}$. There is clearly the ...
- 4,943
1
vote
1
answer
311
views
Probabilistic Johnson-Lindenstrauss Lemma for arbitrary points
Consider the following standard formulation of the Johnson-Lindenstrauss lemma:
Lemma (JL).
For any $0<\epsilon < 1$ and any integer $n$, let $k$ be a positive integer such that $k\geq C\...
- 424
1
vote
1
answer
177
views
Embedding of Two Objects Into Higher Dimensions With Their Sum
Given two vector sets, $\vec x_i$ and $\vec y_i$ (for $i$=1,2,...N, but the dimensionality of each vector can be more than N), let their sum set be $\vec z_i = \vec x_i + \vec y_i$. It's easy to ...
- 1,547
1
vote
1
answer
419
views
Is the direction of the longest line of a polytope unique?
The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions.
The affine subspace is given by:
$X \mbox{ u} = y$
where
$u$ &...
- 121
1
vote
0
answers
27
views
Seeking Help with Classifying Polygons: Waterholes and Airpockets in 2D Space
I am currently in the process of writing software and have encountered a mathematical problem. Perhaps there are some experts here who are familiar with this. It involves the classification of ...
- 11
1
vote
0
answers
43
views
Intersection of unit-norm vectors with a large sum in high dimensions with a spherical cap
Let $d$ and $n$ be integers. For $i \in \lbrace 1,\dots,n \rbrace$ let $x_i \in \mathbb{R}^d$ be a vector such that $\lVert x \rVert=1 $. For a fixed $1/2 < \alpha \leq 1$, assume we have $\lVert \...
- 139
1
vote
0
answers
34
views
Maximum number of concurrencies for $J\cdot L$ hyperplanes in $\mathbb{R}^{J-1}$
I have $J\cdot L$ hyperplanes in $\mathbb{R}^{J-1}$ and want to prove that there cannot be more than $L$ points where $J$ hyperplanes intersect simultaneously (aka. concurrencies).
I suspect that the ...
1
vote
0
answers
109
views
Relation between the dimension of vector spaces and dimension of the space [closed]
Let $A \in \mathrm{GL}(d, \mathbb{R})$ be an irreducible matrix. Assume that $\{V_{n}\}_{n\in \mathbb{N}}$ is a non-zero proper subspace $\mathbb{R}^d$ with dimension $t<d,$ such that $AV_{n}=V_{n+...
- 133
1
vote
0
answers
70
views
Distance to set defined as subzero level set of a continuous function
I am searching for strategies on how to prove/disprove that scalar functions "capture" the distance to the subzero level set of the same function. (Or what topics to study to become better ...
1
vote
0
answers
152
views
Reference request: a class of matrices leading to interesting metric geometry
For $0 \le A \in GL(n,\mathbb{R})$, let $Aw = \Delta(A)$, where $\Delta$ denotes the map taking a matrix to a vector of its diagonal entries and/or forming a diagonal matrix from a vector, according ...
- 15.4k
1
vote
0
answers
1k
views
How to project a matrix to a unitary matrix?
Given a nonzero vector $v \in \mathbb{R}^n$, we all know that it's projection onto the unit $\ell_2$ ball is just $\frac{v}{\|v\|}$. Let $X$ be some nonzero $n \times n$ matrix. What is the projection ...
- 1,703
1
vote
1
answer
95
views
A question on a special "metric"
Suppose we have a function $F: [a,b]^n \to \mathcal{M}_{n \times n }(\mathbb{R})$ where $\mathcal{M}_{n \times n }(\mathbb{R})$ is the space of $n \times n$ real matrices, a compact set $B \subset \...
- 113
0
votes
1
answer
1k
views
Example distance metric that is not conditionally negative definite
Theorem 4.1 of this paper says that there exist distance matrices that are not conditionally negative definite (CND). How do I construct an example of a distance matrix that is not CND? Do you know an ...
- 399
0
votes
1
answer
114
views
Geometric interpretation of a Grammian-like function
Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$:
$$
f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
- 6,351
0
votes
1
answer
445
views
Standard Gram matrices for lattices
I would like to define standard Gram matrices, and use them to help me understand the symmetries of lattices.
I define "standard Gram matrix" as the Gram matrix g that minimizes the ...
- 53
0
votes
1
answer
524
views
Compose/decompose rotation matrix from/to plane of rotation and angle
I would like to compose/decompose an $n$-dimensional orthogonal rotation matrix (restricting to simple planar rotations, which rotates in the specified plane of rotation, and fixes in the plane ...
- 111
0
votes
0
answers
96
views
When can a point be reconstructed from relative angle measurements?
Given a set of points $p_1,\dots,p_n$ in $\mathbb{R}^d$ and a target point $x\in\mathbb{R}^d$, I measure all the angles between all pairs of points and the target point. In other words, I have the ...
- 427
0
votes
0
answers
28
views
Example of a matrix -HDH that is not PSD (with non-euclidean distances D)
It's widely known that, given a matrix of squared Euclidean distances, $\mathbf{D}_{ij} = \| \mathbf{X}_i - \mathbf{X}_j \|^2$, and the centering matrix $\mathbf{H} = \mathbf{I} - \dfrac{1}{n}11^T$, ...
- 101
0
votes
0
answers
55
views
Continuous Functions On Grassmannans under containment restrictions
Let $V$ be a vector space. Suppose that for a $x\in V$, we are given some subspace of dimension no more than d (e.g., the kernel of some operator defined on V, which varies smoothly with x), call it $\...
- 689
0
votes
0
answers
262
views
Lattice basis reductions and finding minimal values
While reading several articles about lattice basis reduction I am left with a few questions.
For one, I came across this piece of text
Let $\alpha$ and $\beta \in \mathbb{R}$. Also let $X>0$ and $...
- 1