0
votes
1answer
27 views

Metric for vector sets

I am currently working on a classification algorithm. Each class is represented by a set of 3D vectors. The cardinality differs for each class. The order of the vectors in a set is completly random. ...
2
votes
1answer
74 views

Is there any proof for this simple observation? [duplicate]

If we consider the Euclidean space $R^3$, it is simply the space where we live. Here we can find only four point such that distance between any two points is a constant. If we consider the Euclidean ...
1
vote
3answers
67 views

Proof of triangle inequality for $d(x,y)=\sqrt{\lvert x-y\rvert}$

There is this problem that says: show that $d(x,y)=\sqrt{\lvert x-y\rvert}$ is a distance function on $\mathbb{R}$, and I am unable to proof the triangle inequality for this? any suggestion I look ...
1
vote
0answers
18 views

Relation between positive definite metric and full basis of a given operator

Let's have some linear space with given indefinite metric. How the fact that metric isn't positive definite is connected with the fact that hermitian (due to the definition of hermicity in a given ...
0
votes
1answer
129 views

Are Euclidean distances a monotone function of inner products?

Does the sum of all pairs of inner-products of k vectors (real) have to decrease if the sums of Euclidean distances between all pairs of $k$ vectors happens to decrease? Similarly-if decrease is ...
-1
votes
1answer
28 views

Finding a condition to make a map a contracting map

Consider $(\mathbb{C}^n ,||\cdot||_{1})$ where $||x||_{1} = \sum\limits_{i=1}^{n} |x_{i}|$ and $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
1
vote
1answer
96 views

If the vector space of all real valued continuous functions on the metric space (X,d) is finite dimensional then X is finite set

If $(X,d)$ is a metric space such that $C(X,R)$ is a finite dimensional real vector space, would any one help me to show that $X$ is finite set? $C(X,R)$ denotes the set of all real valued continuous ...
0
votes
1answer
20 views

Let $a,b \in \mathbb{Z}$ with $a<b$. Determine $ d_n:= | \{ c\in\frac{1}{n}\mathbb{Z} \ | \ a < c < b \} |$

The assignment is: Let $a,b \in \mathbb{Z}$ with $a<b$. For $n\in\mathbb{N}$ determine $ d_n:= | \{ c\in\frac{1}{n}\mathbb{Z} \ | \ a < c < b \} |$. Firstly, I began to see what the ...
1
vote
0answers
87 views

Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
0
votes
3answers
43 views

Prove that this series converges?

I have a Banach space $X$ and a linear operator $A \in L(X)$. $A$ is bounded such that $||A|| <1$. I then have to show that $$log(I-A)=\sum_{n \ge 1} \frac {A^n}n$$ converges. All I can come up ...
0
votes
1answer
50 views

Hypersphere isometry?

I will denote the $n-$sphere of radius $1$ centered at the origin as $\mathbb{S}^n$, so that $$ \mathbb{S}^n = \{ x \in \mathbb{R}^{n+1}\ : \ \|x\| = 1\}. $$ I am stuck on the following problem...I'm ...
1
vote
2answers
80 views

Symmetrical endomorphisms and quadratic forms

(This last part of my linear algebra course is causing me quite a bit of headaches, so please be patient) Let $V$ be a vector space over the real field, and we'll indicate with $(\cdot,\cdot)$ its ...
0
votes
0answers
35 views

Is there an analogue/primitive of PCA which can be in a metric space rather than a vector space?

Principle component analysis PCA is done in a vector space, basically projecting a given vector onto the eigen vectors of the covariance matrix. I'd like to think of a primitive analogoue of PCA, ...
0
votes
1answer
87 views

Set of all matrix of rank $ r $ is open set in $ M_n (\mathbb { R })$

I have no idea how to start it. Actually I have no idea which matrix in $ M_n (\mathbb {R})$ are of rank $ r $. I know all basic result about it. please help me.
2
votes
1answer
103 views

Equivalent codes: Are these two approaches the same?

In the usual theory of codes, a code $C$ of length $n$ of dimension $d$ over a finite field $F$ is a linear subspace $C$ of $\mathbb{F}^n$ of dimension $d$ normed by the Hamming metric. In this sense, ...
1
vote
0answers
61 views

Singular Value Decomposition of Singular Matrices?

$\exists \delta > 0 $ such that whenever $0 < |\varepsilon| < \delta$, $A+\varepsilon I$ is non-singular , for any singular matrix A $\in M_n(\mathbb{C})$ . This is easy to prove. Because ...
0
votes
1answer
54 views

Symmetric Operator with Different dot products

If I have a symmetric operator $A$ in a metric space $\mathscr{M}$. Then $\langle Au,v\rangle =\langle v,Au\rangle $ with the dot product defined in $\mathscr{M}$. My question is, if I keep the same ...
6
votes
2answers
359 views

Cauchy-Schwarz for metrics with arbitrary signatures

When the norm of a vector is always greater than or equal to zero, the Cauchy-Schwarz inequality holds, but what if we look at a metric with an arbitrary signature? Then the inner product of a vector ...
1
vote
0answers
94 views

Reproducing Kernels are Positive Definite. Does the converse hold true?

Does the graph laplacian matrix $L$ form a reproducing kernel- given that the matrix is positive semi-definite. I was told in a hallway by a post doc- a month ago that the pseudo-inverse of $L$ forms ...
1
vote
0answers
464 views

Proof of Isometry: Inner Product Preserving Map

For known points $x_i,x_j,\ldots,x_k$, in $\mathbb{R}^n$, consider a mapping $y_i,y_j,\ldots,y_k$ in $\mathbb{R}^n$ produced by minimizing the function $f(y)=\sum_{i,j} \left \langle x_i,x_j \right ...
6
votes
1answer
363 views

Nonsingularity of Euclidean distance matrix

Let $x_1, \dots, x_k \in \mathbb{R}^n$ be distinct points and let $A$ be the matrix defined by $A_{ij} = d(x_i, x_j)$, where $d$ is the Euclidean distance. Is $A$ always nonsingular? I have a feeling ...
1
vote
1answer
169 views

Maximal color difference

I have a picture consisting of a two-dimensional array of ordered triples (red, green, blue) of real numbers from 0 to 1. I'm looking for something like a norm on pictures which expresses the range of ...
1
vote
1answer
40 views

Recommend a space to analyze the bearing of the vector between any two points

In Euclidean space, given any two points, the vector connecting them can be characterized by length (distance) and direction (bearing). Now I am only interested in the bearing part. And I found it is ...
1
vote
1answer
160 views

Extending $f: X\subset \mathbb{R}^m\to\mathbb{R}^n$ an isometric immersion.

Let $X\subset \mathbb{R}^m$ not empty and $f: X\to\mathbb{R}^n$ an isometric immersion. Prove that there exists an isometric immersion $\varphi: \mathbb{R}^m \to\mathbb{R}^n$ such that ...
2
votes
2answers
310 views

Completeness of normed spaces

As earlier, I have received an answer from this site that Bolzano Weierstrass' theorem is true for finite dimensional normed spaces, but not for infinite dimensional spaces. This, in particular => all ...
5
votes
1answer
296 views

product of hermitian and unitary matrix

Could anyone tell me how to show that, for any $g\in GL_n(\mathbb{C})$, $\exists$ $R$ a hermitian matrix with positive eigenvalues and $U$ an unitary matrix such that $g=RU$? And (I am not sure) can ...
3
votes
2answers
286 views

Open Dense Subset of $M_n(\mathbb{R})$

Well, I know the fact that $GL_n(\mathbb{R})$ is open set in $M_n(\mathbb{R})$, how to show that it is dense also? Well I thought like this: If $A\in M_n(\mathbb{R})$ and If ...
1
vote
1answer
368 views

countable union of proper subspaces

In an interview I was asked to solve a question by using Baire Category Theorem (a complete metric space can not be written as union of nowhere dense subsets), the question was: "Is the vector ...
2
votes
1answer
145 views

A question on norm of error vector

Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...
0
votes
1answer
95 views

Trying to understand how the metric is formed from basis vectors

In the line element $ds=\frac{\partial s}{\partial x^1} dx^1+\frac{\partial s}{\partial x^2} dx^2$ (superscripts are indices, not powers) the basis vectors are defined as $e_1=\frac{\partial ...