The norms tag has no usage guidance.

**23**

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477 views

### Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $||x|| = 1$ and $||Ax|| = ||A||$. The ...

**11**

votes

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118 views

### GPS calculations under $L^p$ norms

GPS calculations require finding a sphere externally tangent to
four given spheres, an
Apollonian problem
in $\mathbb{R}^3$.
The center of that fifth sphere is one of the $16$ possible solutions to
...

**8**

votes

**0**answers

495 views

### Bounding sum of first singular values squared for Kronecker sum of traceless matrices

Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e.
$$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + ...

**6**

votes

**0**answers

130 views

### Norms and distributions

Question 1. Is there a nice or explicit way to describe the class of all distributions (generalized functions) $\mu$ on the $n$-sphere $S^n \subset \mathbb{R}^{n+1}$ for which the function
$$
F(v) := ...

**5**

votes

**0**answers

298 views

### Symmetric matrices with $\rho(A)\gg\|A\|_\infty$

For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...

**5**

votes

**0**answers

183 views

### Reference request: The relationship between norm and trace forms on an Albert algebra

I am interested in either a nice reference, or some clarification.
Overview: I am considering $J_3(\mathbb{O})$, the Jordan algebra of $3\times 3$ self adjoint octonionic matrices. This algebra is a ...

**3**

votes

**0**answers

136 views

### Lenstra's integer programming algorithm: Finding a lattice point “near the center”

I have already posted this question on the mathematics forum, but I suspect the question needs more detailed knowledge than most users have; please excuse the duplicate post. Any help is greatly ...

**3**

votes

**0**answers

130 views

### Norm condition in a Banach lattice

Consider the following "condition (J)" on the norm of a (real or complex) Banach lattice $E$: whenever $x$ and $y$ are disjoint (i.e., $|x|\wedge |y|=0$) then
$\|x+y\|+\|x-y\|=2\|x\|+2\|y\|$.
...

**3**

votes

**0**answers

74 views

### Invexity of the $L_2$ norm

I have the following function:
$ f({\bf A,b}) = \| {\bf y - XAb} \|_2^2$
where ${\bf y}_{n \times 1}$ and ${\bf X}_{n \times p}$ are fixed, and ${\bf A}_{p \times r}$ and ${\bf b}_{r,1}$ are the ...

**3**

votes

**0**answers

133 views

### Diophantine approximations by norms of quadratic irrrationalities

The following problem came up on a mailing list that I subscribe to:
If $\alpha$ is irrational we can find (using continued fractions) infinitely many rational fractions $p/q$ such that $|q \alpha - ...

**3**

votes

**0**answers

352 views

### Bounding linear combination of exponentially decaying functions.

Let $f(x)=e^{-a|x|}$ and $a>0$ a constant. Suppose we take a linear combination
$s(x) = \sum_{i=1}^n \alpha_i f(x-x_i)$
where $\alpha_i\in\mathbb{R}$ and $-r< x_1< \ldots < x_n< r$. ...

**2**

votes

**0**answers

111 views

### Minimize L-infinity norm with restrictions

I need to minimize the following L-infinity norm with respective to $\tau$. L-infinity norm of a matrix $A$ is defined as $\|A\| = max_{i,j}|a_{i,j}|$.
$$
min_{\tau} \| I -S(S+\tau)^{-1}\|
$$
$$
...

**2**

votes

**0**answers

51 views

### Algebraic independence in normed spaces

A set of $n$ points in $\mathbb{R}^2$ is algebraically independent over $\mathbb{Q}$ if there is no polynomial dependency among the $2n$ coordinates.
A result (Lemma 3.3) from "Globally linked pairs ...

**2**

votes

**0**answers

352 views

### Bounds on the effect of a matrix product on the Frobenius norm

I was wondering if there was a way to put upper and lower bounds on the Frobenius norm of a matrix product in relation to the Frobenios norm of one of the individual matrices, i.e,
...

**2**

votes

**0**answers

52 views

### Image of the typenorm contains the squares

I am having a look at the paper Explicit CM-theory for level 2-structures on abelian surfaces by Bröker, Gruenewald and Lauter, and there is an argument which I don't understand. The main reason being ...

**2**

votes

**0**answers

62 views

### Reducing $\ell_1$ norm of non-full-rank matrices

I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ...

**2**

votes

**0**answers

65 views

### König-Wittstock norm on Banach space

Let $(E,\Vert.\Vert)$ be a real Banach space and $\ell\ne 0$
an non-continuous linear form on $E$.
Let $a\in E$ be such that $\ell(a)=1$.
König-Wittstock [Non-equivalent complete norms
and would-be ...

**2**

votes

**0**answers

198 views

### Norm bound of a complex resolvent

A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then
$\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{kk}|-\sum_{j \neq k}|a_{kj}|}$, where the ...

**1**

vote

**0**answers

71 views

### Optimization with random matrix

Consider $J$ a random matrix of size $n\times n$ with i.i.d. Gaussian entries $J_{ij} \sim \mathcal{N}(0,\sigma^2/n)$. Let $f(x)=tanh(x)$, and for $x\in\mathbb{R}^n$, $f(x)$ denotes the vector where ...

**1**

vote

**0**answers

44 views

### Upper semicontinuity in C(X)-algebras. Quotient norm question

upper semicontinuity in C(X)-algebras
In the 5th paragraph of this post, I don't understand why there exists a vector b satisfying $||a+b||_A < ||q_x (a)||_A(x)$
By the definition of the quotient ...

**1**

vote

**0**answers

48 views

### Modified Orthonormal Procrustes Problem

In the general orthonormal Procrustes problem, we want to find an orthonormal matrix $C$ to minimize $\|Y-XC\|_F^2$, where $Y$ is a known $n\times q$ matrix, $X$ is a known $n \times m$ matrix, and ...

**1**

vote

**0**answers

193 views

### RKHS norm and posterior of Gaussian process

In Srinivas et al (2010) [appendix B], the authors claim the following "easy to see" property relating the norm of a function in a RKHS induced by a kernel $k(\cdot,\cdot)$, and its norm in the RKHS ...

**1**

vote

**0**answers

208 views

### Bounding the norm of the Dirichlet kernel as a matrix function

Consider the Dirichlet kerel:
$f(x) = 1+2\sum_{k=1}^{N}\cos(kx)$.
Now, given a diagonalizable real matrix $A$, one can consider $f(A)$, using the standard notation of matrix functions. Namely, $f(A) ...

**1**

vote

**0**answers

123 views

### Interior point optimisation using big M for L1 norm on linear system using Dikin's Affine method

I am a 4th year undergrad surveying student studying computations, specifically $L_{1}$ norm minimisation of residuals in large data sets. To start with (and probably to finish with) I'm using a set ...

**1**

vote

**0**answers

868 views

### What does this notation mean: matrix norm with a two-number subscript

I recently came across this notation, without explanation, in a paper:
$||\mathbf{W}||_{2,1}$
From the context, I know that $\mathbf{W}$ is a matrix, which could be any size, and that ...

**1**

vote

**0**answers

63 views

### Which matrix/operator in a cone has the largest negative spectral part?

Background:
Let $\mathcal{K}$ be set (convex cone, if you like) of symmetric matrices of order $n$. Each matrix $A \in \mathcal{K}$ can be decomposed in a unique way as $A=A_{+}-A_{-}$, where ...

**1**

vote

**0**answers

254 views

### Average weighted value of a linear functional over increasing bounded subsets of Z^n

Say you're working within the finite-dimensional free Z-module $\mathbb{Z}^n$, and you want to impose a "norm" on this module. By a "norm" I mean a function $\|·\|: \mathbb{Z}^n \to \mathbb{R}$ which ...

**1**

vote

**0**answers

160 views

### Norm bound of the entrywise logarithm of a stochastic matrix stationary matrix

Hello,
Denote $\log_\star$ as the entrywise logarithm operation, and let $A$ be some row-stochastic matrix such that $\lim_{p\rightarrow\infty}A^p$ exists and all its entries are non-zero.
As a part ...

**1**

vote

**0**answers

99 views

### expansion with respect to p-norms for p other than 2

Suppose I have an $d$-regular expander graph with $n$ vertices, where the stochastic version of its adjacency matrix $A$ (with entries $1/d$ and zero) has second eigenvalue $\lambda$.
Let $x \in ...

**1**

vote

**0**answers

177 views

### What is the Birkhoff norm of a Perron vector?

Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector?
By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$.
P.S. This is ...

**0**

votes

**0**answers

91 views

### Unit sphere of a norm is a submanifold implies the norm is smooth?

Let us call a norm on $\mathbb{R}^n$ smooth if its restriction $\| \cdot \|:\mathbb{R}^n\setminus \{ 0 \} \to \mathbb{R}$ is a smooth map.
Suppose the unit sphere of a norm $\| \cdot \|$ is an ...

**0**

votes

**0**answers

155 views

### Writing integers in ring of integers of number fields

Given $a,b\in\Bbb N$, we can write $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_ba\rceil$ and $a_i<b<a$.
(1) Supposing if $b\in\mathcal{O}_K$ where $\mathcal{O}_K$ is ring of ...

**0**

votes

**0**answers

96 views

### Does such a function exists that approximates l1 norm by l2 norm for proportional data

A proportional vector $x\in \mathbb{R}_{+}^{d}$ is defined as (1) $x_{i}\geq 0$ ;(2) $\sum_{i=1}^{d}x_{i}=1$; $d$ is independent of anything else.
For two arbitrary proportional vector $x$ and $y$, ...

**0**

votes

**0**answers

81 views

### The 2-norm of a positive circulant matrix

Define a circulant matrix $A$ for complex numbers $a_1, a_2, ..., a_n$ as follows:
$$
\text{circ}(a_1,\ldots,a_n)=
\left[ \begin{matrix}
a_1& a_2 & \cdots & a_{n-1} & a_n \\
a_n& ...

**0**

votes

**0**answers

43 views

### Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions
Let
$L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric),
$a,b$ be arbitrary $n$-dimensional points,
$c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...

**0**

votes

**0**answers

79 views

### Characterize the set of roots of cubics with certain properties

Let $P(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $3$. Suppose that $\alpha_1, \alpha_2, \alpha_3$ are roots of $P(x)$. For what such $P(x)$ is it the case that the ring of integers ...

**0**

votes

**0**answers

90 views

### “Almost orthogonalizing” matrices using a signature matrix

Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs).
Then $||AB||_{op} \leq ...

**0**

votes

**0**answers

291 views

### Closed-form expressions for dual norms of real normed vector spaces

Didn't get any biters over at MSE, so I figure this place might be more appropriate...
Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by ...

**0**

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**0**answers

358 views

### Surjectively isometric normed spaces: Hamel vs (extended) Schauder dimension

Bonjour/bonsoir à toutes et à tous.
This may really be a very basic question, but... Let $\mathbf{X} \equiv (X, \|\cdot\|_X)$ and $\mathbf{Y} \equiv (Y, \|\cdot\|_Y)$ be surjectively isometric (1) ...