All Questions
7,280 questions
3
votes
2
answers
1k
views
"Main" diagonal of a matrix
Hello!
I'm in search of some (possibly statistical) measure for matrices. I want to classify a square matrix as having the largest numbers running along the main diagonal or along the anitdiagonal. ...
- 17.9k
7
votes
1
answer
1k
views
Is the absolutely continuous image of a nowhere dense set is also nowhere dense?
Let $f: [a, b] \subseteq \mathbb{R} \to \mathbb{R}$ be an absolutely continuous map. Does $f$ map a nowhere dense subset of $[a, b]$ to a nowhere dense set?
Remarks:
The answer is "no" if $f$ is ...
- 66
22
votes
1
answer
13k
views
Non-diagonalizable complex symmetric matrix
This is a question in elementary linear algebra, though I hope it's not so trivial to be closed.
Real symmetric matrices, complex hermitian matrices, unitary matrices, and complex matrices with ...
- 156k
0
votes
1
answer
406
views
Operation of GL_n(Z/bZ) [closed]
I want to show, that $GL_n(\mathbb{Z}/b\mathbb{Z})$ operates transitively on
$X = \{ (v_1, \ldots, v_n) \in (\mathbb{Z}/b\mathbb{Z})^n \ | \ v_1\mathbb{Z}/b\mathbb{Z} + \ldots + v_n\mathbb{Z}/b\...
- 65.4k
9
votes
2
answers
804
views
Partition of R into midpoint convex sets
We say that a subset $X$ of $\mathbb{R}$ is midpoint convex if for any two points $a,b\in X$ the midpoint $\frac{a+b}{2}$ also lies in $X$.
My question is: is it possible to partition $\mathbb{R}$ ...
- 6,224
1
vote
0
answers
1k
views
Covariance matrix formula interpretation - what am I missing?
I'm reading a paper that outlines the calculation of a covariance matrix like the following:
$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$
What is the order of this matrix? My interpretation ...
- 111
2
votes
3
answers
946
views
How can I measure the Morse index in infinite dimensions?
Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$...
CommunityBot
- 1
2
votes
0
answers
354
views
What is this effect in Fourier/additive synthesis called?
Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was $\sum a_{n} sin(2 \pi x * n)$ and its frequencies were $n*f_{...
- 165
2
votes
3
answers
772
views
Matrix decomposition problem
Given a pair of distributions $x,y\in(0,1]^{n\times 1}$, so that $1^Tx=1$ and $1^Ty=1$,
I want to build a matrix $C$ (change matrix) that satisfy at least the following properties:
i) $C$ is ...
- 51
5
votes
0
answers
558
views
continuous selection of a multivalued function?
The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...
- 1,839
10
votes
2
answers
3k
views
Continuous function from $[0,1]$ to $[0,1]$
Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?
- 4,736
2
votes
2
answers
317
views
Bibliography for topologies defined by a family of seminorms
Hello
I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource.
Thank you very much.
- 27.7k
2
votes
1
answer
897
views
Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis
Greetings,
I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar ...
- 50.6k
6
votes
7
answers
5k
views
Best way to teach concept of real numbers using a hands-on activity?
I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.
- 45.7k
2
votes
2
answers
535
views
Sequence of constant rank matrices
Let $A_k$ be a sequence of real, rank $r$, $n$ x $m$ matrices such that $A_k$ converges to a rank $r$ matrix $A$. Let $v_k, u_k$ be sequences of vectors such that $u_k\rightarrow u$ and $A_k v_k=u_k$. ...
- 32.4k
3
votes
1
answer
538
views
Non-negative matrices with prescribed Perron-Frobenius eigenvectors
In my research I came across the following question.
Let $A$ be an integer non-negative matrix (every entry of $A$ is non-negative) and $x = (x_1,...,x_n)^T$ the probability Perron-Frobenius ...
- 3,929
2
votes
0
answers
517
views
When deRham curve is bijection?
Motivation: Suppose we have deRham curve. From wikipedia:
Consider some metric space $(M,d)$ (generally $R^2$ with the usual euclidean distance), and a pair of contraction mappings on M:
$d_0:\ M \...
- 65.4k
4
votes
2
answers
340
views
Embeddings of Weighted Banach Spaces
Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces
$$ \Omega_p:=\left\{ x\in \Omega\left| \left[\sum_{i\in\mathbb{Z}^d}\frac{|x_i|^...
- 25.8k
10
votes
0
answers
439
views
Evaluating Shintani cone zeta functions
Hi everyone
I am trying the evaluate sums of the form
$$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$
...
- 47.3k
6
votes
3
answers
1k
views
Dependence of error on mesh for Riemann sums
Suppose $f$ is continuous on $[a,b]$ with $I = \int_a^b f(x)\: dx$,
and for every $\epsilon > 0$ let $\delta(\epsilon)$ be the largest
$\delta > 0$ such that every Riemann sum arising from a ...
- 15.4k
2
votes
2
answers
3k
views
Statement of Lagrange's theorem on determinants(elementary question).
Apologies for this elementary question; but I was unable to find a reference otherwise.
Let $A, B, C$ be square matrices of the same dimension. Then,
$$\begin{vmatrix} A & C \\\ 0 & B \end{...
- 33.8k
5
votes
1
answer
2k
views
Self-similar matrices? [closed]
Does anyone know anything about self-similar (infinite) matrices, with more or less fractal(-like) structure and admitting meaningful matrix-algebra operations?
- 7,800
2
votes
1
answer
1k
views
How can we use the bounded convergence theorem in this proof of the Riesz Representation Theorem?
I'm studying the proof of the Riesz Representation Theorem as it appears in Ch. 6 of Royden's Real Analysis. When I looked on the web I noted there are a few different theorems that go by the name "...
- 61.9k
1
vote
2
answers
923
views
Extremum under variations of a traceless matrix
Sorry for my precedent tentative, I was a little hasty:
Ok, I think I'd better put the original problem:
I have an action of three fields: $A$ which is the spin-connection, $B$ an skew-symmetric 2-...
- 33.3k
11
votes
1
answer
410
views
An "existence contra partition of unity" statement for integer matrices?
While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind.
Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...
- 2,600
5
votes
5
answers
2k
views
Cardinality of Equivalence Classes of Cauchy Sequences
What's the cardinality of a single equivalence class of Cauchy sequences in ℚ?
To clarify, I'm not asking for the cardinality of the real numbers, but for the cardinality of the set of Cauchy ...
- 236k
4
votes
2
answers
730
views
Decomposition of Hölder continuous functions
Let $\alpha\in(0,1)$ and $\eta\in\Lambda_0^\alpha(\mathbb{R})$ be a compactly supported Hölder continuous function of order $\alpha$. I would like to show that, for any $n\in\mathbb{N}$, it is ...
- 1,981
3
votes
1
answer
263
views
Asymptotically multiplicative functions and matrices
Hi,
Let $\mathbb{N}_{cop}^2$ denote the set of all pairs of coprime natural numbers. A function $f:\mathbb{C}\rightarrow\mathbb{C}$ is called asymptotically multiplicative, iff $\epsilon_{m,n}:=f(mn)...
- 25.8k
5
votes
5
answers
5k
views
Notions of Matrix Differentiation
There are a few standard notions of matrix derivatives, e.g.
If f is a function defined on the entries of a matrix A, then one can talk about the matrix of partial derivatives of f.
If the entries of ...
- 2,284
1
vote
4
answers
385
views
Is it that only with normal matrices, the transition matrix to its [del: inherent] [ins: own] basis is unitary?
Does this even make sense what I translated into english?
PS. I am probably gonna delete this question eventually
- 13