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Questions tagged [norms]

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1
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1answer
91 views

Characterizing a norm on sequences

Let $\{a_i\}$ be a sequence of reals such that $|a_i|\geq|a_{i+1}|$ for all $i$, and consider the following norm: $$\|\{a_i\}\| = \sup_k \frac{1}{\sqrt{k}}\sum_{i=1}^k |a_i|~.$$ One can see that -- ...
6
votes
2answers
191 views

Proving an infinite norm minimization problem has finite support (non-convex p-norms)

Consider an optimization problem over infinite variables: $$ \begin{align} \min_{x}~& {\left\lVert{x}\right\rVert }_p \\ \text{s.t}~& \left\langle x, a_n\right\rangle \ge 1~,~\forall n=1,\...
0
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1answer
58 views

Total variation norm estimate

I have the following question concerning an estimate of the total variation norm. Let $\mu$ be a bounded Borel measure on $\mathbb{R}$ and denote by $\mu_t$ the measure defined by $\mu_t(\Omega):=\mu(\...
0
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0answers
15 views

How to derive a bound of distortion / error between two different tensor decompositions

Consider a tensor $\mathcal{X}\in\mathbb{R}^{I\times J\times K}$. It can be approximately decomposed/factored in multiple ways. Namely by using the TUCKER3 decomposition: $\mathcal{X}\approx \sum_{p=...
0
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1answer
193 views

Does this norm have a specific name? Banach space? References?

Let $(X,\mathscr{B},\mu)$ be a $\sigma$-finite measure space. Let $\gamma$ be a probability measure on $L_2(\mu)$ with $\mathrm{supp} \, \gamma = L_2(\mu)$ and existing first moment. Then $$ f \...
6
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0answers
160 views

Find a condition such that the spectral radius for a special matrix is smaller than 1 (or a matrix norm smaller than 1)

We need a help to find a reasonable condition such that the spectral radius for a special matrix $\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}$ is smaller ...
3
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1answer
77 views

Largest eigenvalue of product of orthogonal-projection rank-1 perturbation

Suppose I have a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ with $n$ linearly indepedent columns $a_1,...a_n$ in $\mathbb{R}^n$. All columns $a_i$ has norm 1, but they are not ...
0
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2answers
115 views

Existence of p=-infinity norm [closed]

Given a vector $\mathbf {x} =(x_{1},\ldots ,x_{n})$ the p-norm is defined as $$ \left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p} $$ for $p \geq 1$. For $...
3
votes
1answer
81 views

Sequence in *-algebra with different limits for two C*-norms?

The following question looks simple, but the answer is not obvious for me: Let $S$ be a $*$-algebra and $\left\Vert \cdot \right\Vert _{1}$, $\left\Vert \cdot \right\Vert _{2}$ $C^*$-norms on $S$ ...
1
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1answer
84 views

Is continuity preserved under norm operations

Let $F$ be a continuously differenable function over $\mathbb{R}$. Let $\Omega$ be a bounded subset of $\mathbb{R}^2$. Assume that for every $w\in L^2(\Omega)$ then $v(x)=F(w(x))$, $x\in \Omega$, is ...
0
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1answer
50 views

Quasiconvexity property of quasinorms

Schatten $p$ norm is convex when $p\geq1$ holds and if $p\in(0,1)$ it is quasinorm. If $p\in(0,1)$ then is Schatten $p$ norm quasi convex? I am interested in definition of quasi convexity here https:...
5
votes
1answer
119 views

Image of the norm map for degree $3$ galois extension over $\mathbb{Q}$

I want to construct a cyclic division algebra of degree $3$ over some degree $3$ Galois extension $E$ of $\mathbb{Q}$. So the construction is as follows: As a set $D=E\oplus uE \oplus u^2 E$ where $u$...
17
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2answers
423 views

On a special type of normed linear spaces

Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying $$ \|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z} $$ is a group ...
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0answers
48 views

Unclear inequality of L2 norms (Poisson equation for modeling flow)

I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...
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0answers
58 views

Show that the norm's bound is an exponent

Let $f$ be a continuous function on $S^2$. Consider $g\in C^{\infty}(R)$, such that $g(x)=1$ for $|x|\leq 1$ and for $|x|\geq 2$. Let $h(x)=g(x)-g(2x)$. The notation $proj_k$ denotes the orthogonal ...
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0answers
44 views

Approximating norms using numerical integration? [closed]

I have a sequence of functions $u_m$ in $H^1(\Omega)$, where $\Omega$ is Lipschitz such that $u_m(x)=\int_{|y|\le \epsilon} \, f_m(x,y) \, dy$, but the integral cannot be expressed in terms of ...
-1
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1answer
72 views

Bound for psd-matrix weighted norm of two related vectors

Let two vectors, $\mathbf{x}, \mathbf{y}$ be related as: $0 \leq x_i \leq \lambda y_i$, for some $\lambda > 0$. That is, $\mathbf{x}$ is coodinate-wise dominated by a scaled version of $\mathbf{y}$....
2
votes
1answer
165 views

Regarding spectral radius

Let $A$ be a $C^*$ algebra. Let $a\in A$ be such that $a^*a-aa^*\geq 0$. Doe this imply that the spectral radius of $a$ is equal to $\|a\|$?
0
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1answer
74 views

On an error bound for matrix constraints

Given $A,B\in\mathbb R^{n\times n}$ such that there is an $U\in\mathbb R^{n\times n}$ with $UU'=I$ and $UAU'=B$. Suppose each entry of $V$ is within $\pm\epsilon$ of each entry of $U$ is there a way ...
5
votes
2answers
579 views

Nuclear norm as minimum of Frobenius norm product

Nuclear, or trace, or Ky Fan, norm of a matrix is defined as the sum of the singular values of the matrix. It is claimed that $$ \|X\|_\sigma = \min_{UV^T=X} \|U\|\|V\| = \min_{UV^T=X} \frac{1}{2}(\|...
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0answers
115 views

Convexity of the Frobenius norm of the product of matrices

I have a question similar to Convexity of the Frobenius norm of the product of two matrices. I am not able to comment on that question as I don't have enough reputation, and that is why I have asked a ...
0
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0answers
47 views

$H_\infty$-norm of a time delay system

I have a linear dynamical stable delayed system as follows, $\dot{x}=Ax(t-\tau)+Bu$, where $A_{n \times n}$ is a stable matrix with all its eigenvalues located on the open left half of complex plane. ...
0
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0answers
42 views

$H_2$-norm of a time delay system

I have a linear dynamical stable delayed system as follows, $\dot{x}=Ax(t−τ)+Bu$, where $A_{n \times n}$ is a stable matrix with all its eigenvalues located on the open left half of complex plane. I ...
9
votes
1answer
243 views

Regular $p$-norm of a matrix

Let $n \in \mathbb{N}$ and $p \in [1,\infty]$ be fixed and endow $\mathbb{C}^n$ with the $p$-norm $\|\cdot\|_p$. For every matrix $A \in \mathbb{C}^{n \times n}$ we denote the operator norm of $A$ as ...
1
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1answer
209 views

Which norms on vectors can be consistently decomposed?

I need to know which permutation-invariant norms can be consistently decomposed in the sense that for any vector $v = (a,b,c)$ we have that $$\|(a,b,c)\| = \|(\|(a,b)\|,c)\|.$$ More precisely, let $v ...
3
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0answers
116 views

The inverse of sum of two positive matrices with almost orthogonal supports

I am interested to find an approximate formula for $$A (A+B)^{-1} A\ ,$$ for two positive matrices $A$ and $B$ whose supports are almost orthogonal. If the support of $A$ and $B$ are orthogonal ...
1
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1answer
78 views

Nonlinear low-rank approximation - corrected

I would like to state that this is related to a past question of mine which contained errors and now appears in the corrected form, with the erroneous one deleted and closed. In my research of linear ...
1
vote
1answer
77 views

Possible analytical way to solve or approximate a specific optimization problem's solution

In my research on linear algebra and optimization, I have come across the following problem repeatedly: Given constant matrices $C\in\mathbb{R}^{k \times k}$ and $X\in\mathbb{R}^{n \times n}$, $$\...
0
votes
1answer
100 views

Modification of a known optimization problem

In my research of linear algebra and optimization, I wish to modify the following well-known problem: $ \min \lVert x-Ax \rVert$ subject to $ rank(A)\leq k $ where $ x $ is a given column vector ...
6
votes
1answer
129 views

The Euclidean norm and $k$ largest elements

This is not a homework problem, although I fear it may turn out to be at that level. For any nonnegative $x\in\mathbb{R}^n$, let $f_k(x)$ be the sum of the $k$ largest values in $x$, and define $$f(x)...
1
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1answer
169 views

Do we have that $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}$ and $\| u \|_{W^{2,p}(\Bbb R^d)}$ are equivalent norms?

Is it possible that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}~~~~and~~\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms? This results is pretty easy and straightforward for $p=2$ ...
0
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0answers
83 views

Is this possible to calculate norm of a vector using its inner product with another vector

Given a manifold $M$ with two Riemannian metrics $g_1$ and $g_2$ on $M$, a vector field $\xi$, and a smooth function $f:M\to \mathbb{R}$, is this possible to calculate $g_2(\xi,\xi)$ having the ...
2
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1answer
154 views

A priori estimate of an inhomogeneous p-Laplace equation with Dirichlet boundary condition

I'm currently working on this Dirichlet problem: \begin{cases} div(\sigma |\nabla u|^{p-2} \nabla u) = f &\quad {in }~ \Omega\\ u = g &\quad in~\partial\Omega \end{cases} with $\sigma \in L^...
11
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3answers
849 views

What are the matrices preserving the $\ell^1$-norm?

So I am inspired by unitary matrices which preserve the $\ell^2$-norm of all vectors, so in particular the unit norm vectors. But then I saw that the $\ell^1$-norm of probability vectors is preserved ...
2
votes
1answer
167 views

Differentials in different topologies

I have read (In French)that the differential of a function depends on the topology and not the norm, the latter is rather easy to grasp, the first is hard for me to construct. Norms being equivalent ...
1
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1answer
177 views

Norms of elements in Artin-Schreier extensions

The following is claimed in the proof of Theorem 7.5 of Auslander, Goldman, "The Brauer group of a commutative ring": Let $k$ be a nonperfect field of positive characteristic $p$, let $K := k(x)$ ...
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0answers
72 views

Operator norm for $\max\frac{\Vert x \Vert _1}{\sqrt {x'Cx}}$

Suppose $C$ is a $n$ by $n$ real symmetric matrix, and $x\in R^n$. Is there an operator norm of $C$ for $\max\frac{\Vert x \Vert _1}{\sqrt {x'Cx}}$? If I decompose $C$ into $A'A = C^{-1}$, It seems ...
1
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1answer
220 views

Is there a standard proof that the L^1 norm > constant * sup norm for functions with derivative bounded above by K on the unit disk in R^n?

Suppose that you have a bounded function $f(x)$ on a compact domain in $\mathbb{R}^n$. It's easy to see from Holder's inequality that $$ ||f||_1 \leq \operatorname{Volume}(D) ||f||_\infty. $$ There ...
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1answer
110 views

Is there a commonly used short name for “squared Euclidean distance”? [closed]

In an optimization program I pass around distance values quite often. In my case these are simple 2D Euclidean distances $\sqrt{\Delta x^2+\Delta y^2}$. Since I want to perform the square root ...
6
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0answers
233 views

Upper bound for $\|\textbf{D}^{-1}\|$, where $\textbf{D}$ is a matrix with specific sparse pattern

Consider the block matrix given by $$\textbf{D} = \left[ \begin{array}{ccc} \left[ \begin{array}{ccc} D & \ldots & D\\ \vdots & \ddots & \vdots\\ D &...
3
votes
1answer
242 views

Norm and trace inequalities

If $A$ and $B$ are two positive definite matrices such that $\|A\| \leq \|B\|$ for every unitarily invariant norm $\| \cdot \|$, and $U$ is an $n\times k$ matrix with adjoint $V$ such that $VU = I_k$, ...
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0answers
122 views

Strong smoothness of Lp norm

I asked this question in math.stackexchange but got no answers (link: https://math.stackexchange.com/questions/2323520/strong-smoothness-of-lp-norm). So I decided to ask this question here. Hope I ...
2
votes
1answer
155 views

Tighest Gap $\|x\|_1/\|x\|_2$ between $\ell^1$ and $\ell^2$ norms

I'm looking specifically at the optimization problem $$ \begin{align*} \text{maximize: }& n - \frac{\|\lambda\|_1^2}{\|\lambda\|_2^2}\\ \text{subj. to: }& \lambda \succeq \epsilon\mathbf{1} \...
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0answers
82 views

Relation between Holder norm and $p$- variation norm

Consider a measurable function $f:[a,b]\to\Bbb R^d$. For $0<\alpha\le1$ we define the $\alpha$-Holder norm as $$ ||f||_{\alpha,[a,b]}:=\sup_{a\le s<t\le b}\frac{|f(t)-f(s)|}{|t-s|^{\alpha}} $$ ...
5
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1answer
146 views

Are there primitive quartic CM fields whose norms of units give all totally positive units of the real quadratic subfield?

Let $K$ be a primitive (i.e. not biquadratic) quartic CM-field. That is we have $[K:\mathbb{Q}]=4$ and let $K_0=\mathbb{Q}(\sqrt{d})$ be the totally real quadratic subfield, here $d> 1$, that is we ...
0
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2answers
143 views

How to prove that a subset in a vector space is an indicatrix?

Suppose that $\pi:(V_1,F_1)\to V_2$ is a linear surjective map, where $V_1$ and $V_2$ are vector spaces and $F_1$ is a Minkowski norm on $V_1$. Let $B_1$ be the unitary ball on $V_1$. Define $B_2:=\pi(...
4
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1answer
233 views

Conditions for the support function of ellipsoid to define a norm

Let $C$ be a (nonempty) convex compact subset of $\mathbb R^n$. General question: Under what conditions on $C$ does the support function $$\sigma_C(x) := \sup_{y \in C}x^Ty $$ define a norm on $\...
1
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0answers
472 views

Is Hilbert–Schmidt and Frobenius norm the same?

From the definition on R those two norm are the same: http://mathworld.wolfram.com/FrobeniusNorm.html, http://mathworld.wolfram.com/Hilbert-SchmidtNorm.html Is there some difference (on C) or ...
1
vote
0answers
76 views

Question on norms on tensor product and algebra

Question 1 Let $V,W$ be normed spaces and $\iota:V\times W \rightarrow V\otimes W$ be the canonical (algebraic) bilinear map. It can be easily shown that for any normed space $X$ and a continuous ...
5
votes
2answers
168 views

On the coincidence (or non-coincidence) of two norms defined on the quotient of a given Hilbert $ C^{\ast} $-module by a certain linear subspace

Let $ A $ be a $ C^{\ast} $-algebra, $ I $ a closed two-sided ideal of $ A $, and $ \mathcal{E} $ a Hilbert $ A $-module. Let $$ \mathcal{E}_{I} \stackrel{\text{df}}{=} \{ x \in \mathcal{E} \mid \...