# Tagged Questions

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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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.
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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