Tagged Questions
1
vote
0answers
24 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
28 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 ...
2
votes
1answer
70 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
71 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
262 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
217 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
127 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
32 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
138 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
162 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
175 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
160 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
190 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
136 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
76 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 ...
