The norms tag has no usage guidance.

**10**

votes

**3**answers

519 views

### Are norms intrinsically $\mathbb{R}$-valued?

Another way of phrasing this: are there any viable definitions of something which is norm-like but whose range is in a linearly ordered rig (for example) rather than $\mathbb{R}$?
I have searched a ...

**6**

votes

**2**answers

334 views

### $\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$

Let $T$ be a linear operator acting on a finite-dimensional real or complex
vector space. As a direct consequence (or rather a particular case) of the
Riesz-Thorin theorem, we have
$$ \|T\|_2 \le ...

**2**

votes

**1**answer

98 views

### Bound on the 2-norm of a “special” matrix

Let $S\in\mathbb{R}^{n\times n}$ be such that $\|S\|_2\leq 1$, $P\in\mathbb{R}^{n\times m}$, $m<n$, with orthogonal columns ($P^TP=I$) so that $PP^T$ and $I-PP^T$ are orthogonal projectors, and ...

**10**

votes

**1**answer

759 views

### On bi-invariant metrics on groups

I'm looking for a quick, snap-your-fingers proof of the following result:
A continuous length metric on $\mathbb{R}^n$ that is invariant under translations comes from a norm.
To be clear about ...

**3**

votes

**1**answer

313 views

### Estimate infinity norm with Lp and W1p norm

Let $p \in [1,\infty)$. Does there exist $C>0$ such that for every $f \in W^{1,p}([0,1],\mathbb{R})$ we have
$$\|f\|_{L^\infty}\leq C\|f\|_{L^p}^{1-\frac{1}{p}}\|f\|_{W^{1,p}}^{\frac{1}{p}}?$$
My ...

**5**

votes

**0**answers

289 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 ...

**2**

votes

**0**answers

189 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 ...

**5**

votes

**1**answer

291 views

### An inequality involving the spectral norm of a complex matrix

Let $A,B \in {M_n}(R)$ be real $n \times n$ matrices and let matrices $|A|$ and $|B|$ contain the absolute values of the elements of $A$ and $B$ respectively. Construct the complex matrices $C = A + j ...

**17**

votes

**2**answers

734 views

### An equivalence relation for norms

Let us say that two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a real vector space $V$ are strongly equivalent if there exists a constant $\lambda \geq 1$ such that
$$
\frac{1}{\lambda} \left( \|x\|_1 ...

**4**

votes

**1**answer

132 views

### Estimating the probability that $\|Av\| \ge \|v\|$

Given a diagonalizable matrix $A \in \mathbb{R}^{n \times n}$ with real eigenvalues, satisfying $1+c_1 \le \rho(A) \le 1+c_2$ $(0<c_1 \le c_2)$, obviously there exists a $v \in \mathbb{R}^{n}$ such ...

**1**

vote

**0**answers

195 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) ...

**6**

votes

**0**answers

129 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) := ...

**3**

votes

**1**answer

400 views

### About generalized Minkowski inequality

For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality
$f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + ...

**11**

votes

**2**answers

1k views

### Hölder's inequality for matrices

I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if
$$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$
But ...

**3**

votes

**1**answer

134 views

### Is this function of a matrix convex?

Let $\mathcal{N}_{n}$ be the set of symmetric nonnegative irreducible matrices. For a matrix $A \in \mathcal{N}_{n}$ let $v^{A}$ be its Perron vector, normalized so that $||v^{A}||_{2}=1$.
Define the ...

**0**

votes

**1**answer

679 views

### How to determine the distance between two matrices under the meaning of a matrix function? [closed]

Suppose a nonlinear infinitely continous differentiable function $f:\mathbb{D}\mapsto \mathbb{R^+}$, where $\mathbb{D}\subset\left\{X|\text{rank}{X}=2,X\in\mathbb{R}^{3\times 3}\right\}$ is a ...

**2**

votes

**1**answer

81 views

### Relating joint probability to norm of vector of probabilities

I have a set of Bernoulli random variables $X_1,X_2,\ldots, X_n$ and I would want to bound the joint probability $P(X_1=0,X_2=0,\ldots, X_n=0)$ using the norm $\lvert \lvert \mathbf{p}\rvert \rvert$, ...

**4**

votes

**1**answer

182 views

### A homogeneous but slightly asymmetric inequality

I need to prove the following inequality: for any $Z=(z_1,\dots,z_l)\in\mathbb{C}^l$ for any $p\geq 2$ and $l\geq 2$
\begin{equation}
\left|\left|\sum_{j=1}^l ...

**0**

votes

**1**answer

109 views

### Norm bound on eigen-vector change caused by rank-one update

Suppose $A$ is a positive semi-definite, Hermitian matrix with a unit-norm eigen vector $\textbf{v}$ corresponding to its largest eigen value $\lambda$. Let $B = A + \alpha \textbf{z}\textbf{z}^H$, ...

**0**

votes

**0**answers

87 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 ...

**4**

votes

**1**answer

83 views

### Norms of B-spline coefficients

In Shumaker's book (Spline Functions: Basic Theory), we know that the $l^\infty$-norm of B-spline coefficients is bounded above and below by the $L^\infty$-norm of the spline itself. Are there similar ...

**0**

votes

**0**answers

273 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 ...

**1**

vote

**0**answers

120 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 ...

**12**

votes

**1**answer

899 views

### A property that forces the NORM to be induced by an INNER PRODUCT

Let $(E, \|\cdot\|)$ be a real normed vector space such that for any $a,b\in E$,
$$ \|x +y\|^2 + \|x-y\|^2 \geq 4 \|x\|\cdot \|y\| $$
I want to show that the norm is induced by an inner product. Any ...

**3**

votes

**0**answers

132 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 - ...

**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 ...

**1**

vote

**2**answers

191 views

### Matrix-Norm aquivalence with p-Norm [closed]

Let $A$ be a square Matrix and $||\cdot ||_p$ the induced Matrixnorm for $1 \leq p \leq \infty$. Is it true that
$$||A||_p\leq \max(||A||_1,||A||_{\infty})?$$
For $p=2$ the answer is yes because ...

**6**

votes

**1**answer

445 views

### Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?

The question is in the title: Let $E$ be an origin-centered ellipse in ${\mathbb R}^2$ and let $S$ be an "$L^p$-circle": $S = \{(x,y) : |x|^p + |y|^p = \text{const}\}$, where $1 \leq p \leq \infty$. ...

**1**

vote

**0**answers

797 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 ...

**0**

votes

**1**answer

355 views

### solving trace norm equality [closed]

Problem Formulation
under what conditions can we solve $\mathrm{trace}(\mathbf{AB})=0$ ? or more specifically, when will $\mathrm{trace}(\mathbf{AB})=0$ implies that $\mathrm{trace}(\mathbf{B})=0$.
...

**2**

votes

**1**answer

296 views

### Extension of equivalent norms

Let $(X,||\cdot||_1)$ be a normed space and $Y$ a linear subspace of $X$. Let $||\cdot||_2$ be a norm on $X$ which is equivalent to $||\cdot||_1$ on $Y$. Does there exist a norm on $X$ that coincides ...

**3**

votes

**1**answer

350 views

### Matrix norms / eigenvalues / singular values / another thing

OK, here is what is probably a stupid question.
Let $M$ be a non-symmetric real matrix: for example, the shear matrix
$\left( \begin{array}{cc} 1 & 1 \\\ 0 & 1 \end{array} \right)$.
There ...

**1**

vote

**0**answers

62 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 ...

**5**

votes

**1**answer

286 views

### The minimal norm of a shifted stochastic matrix

Hello,
Given a row-stochastic matrix $M$ with singular values $\sigma_{1}\geq\ldots\geq\sigma_{n}$, I am looking for an upper bound on the expression: $\min_{\alpha}\parallel M- ...

**2**

votes

**2**answers

335 views

### On matrix norms

It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:
$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.
Suppose we define a different function of ...

**0**

votes

**1**answer

389 views

### Norms agreeing on dense subspace [closed]

Suppose $(B,\|\cdot\|)$ is a Banach space, $V\subset B$ a dense subspace, and $V$ is equipped with a norm $\|\cdot\|_V$ such that $\|x\|_V = \|x\|$ for all $x\in V$.
Is $(B,\|\cdot\|)$ a completion ...

**0**

votes

**1**answer

243 views

### Absolute norms and 1-unconditional sums

Absolute norm
Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that
$$
\|(x,y)\|_N=N((\|x\|, ...

**0**

votes

**2**answers

734 views

### Inequality of Lebesgue integral with $L^p$-norm

Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$.
I ...

**1**

vote

**0**answers

253 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 ...

**0**

votes

**1**answer

296 views

### Integral inequality

Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic ...

**1**

vote

**0**answers

158 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

**3**answers

1k views

### Comparison of the L_p norm of a matrix and its entry-wise absolute value

Suppose $A_{n \times n}$ is a matrix and $A' = (|A_{ij}|)$ is its entry wise absolute form, can be give an upper bound and lower bound of the L_p norm $\|A\|_p$ using the L_p norm of the absolute ...

**8**

votes

**2**answers

312 views

### Projections onto $n$-codimensional subspaces of a Banach space: norms.

Hello, I'd like some help to find an answer I've been looking for since this morning.
Let $X$ be a Banach space and let $Y$ be an $n$-codimensional subspace of $X$. Let $P$ be a projection from $X$ ...

**3**

votes

**1**answer

822 views

### norm of (sub)stochastic matrix

Is there any bounds for the norm of sub-stochastic matrix? (But it's not doubly stochastic matrix, I mean only the row sum is less than 1, while the column sum may not.

**1**

vote

**0**answers

97 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 ...

**2**

votes

**2**answers

621 views

### How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional Lp-normed vector space?

Say you have a finite-dimensional vector space $V$ with an $L^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $L^p$ norm, so the unit sphere in $V_s$ ...

**1**

vote

**0**answers

174 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 ...

**2**

votes

**2**answers

2k views

### Dual Norm For Sum of 2-Norms

What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$
$\|\mathbf{x}\| = ...

**1**

vote

**2**answers

288 views

### An inequality with $\ell_p$ norm

I encounter the following claim in my research for which I couldn't get a solution for a long time. I asked a more general version of the question at math.stackexchange which did not attract much ...

**3**

votes

**3**answers

1k views

### upper bounds on a certain matrix norm

Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?