Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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Is it true that $\|M(I-M)^{-1}\|_{\infty} \leq \frac{\|M\|_{\infty}}{1-\|M\|_{\infty}}\,?$

I've encountered a non-symmetric matrix $M$ with nonnegative elements ($M_{ij} \geq 0$), satisfying $\|M\|_{\infty} < 1$, which I need to bound. Is it true that $$ \|M(I-M)^{-1}\|_{\infty} \leq ...
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57 views

A condition equivalent to orthogonality

Prove that in any inner product space: $x$ and $y$ are orthogonal if and only if $||x+\alpha y||\ge ||x||$ for every scalar $\alpha$.
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43 views

Spectral norm upper bound for covariance matrix

Let $\|\cdot\|_2$ be the spectral norm. Let $x_1,\dots,x_n$ be i.i.d. draws from $N(0,S)$. Let $\lambda_1,\dots,\lambda_n$ be some real numbers. Is it true that $$\|\sum_{i=1}^n \lambda_i x_i ...
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59 views

Relation between Frobenius norm and eigenvalues

I'm considering a stochastic multivariate process, the stability of which implies that all eigenvalues $\lambda_i$, $i = \overline{1,n}$ of a certain square real-valued matrix $A$ lie within the ...
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27 views

Bound a Lyapunov storage function

How to effectively bound the following entity to deduce its definite negativeness $\dot{v} = -k_1 e_1^\top A e_1 + k_1 e_1^{\top} A e_2- k_2|e_2|^2$, with A a positive definite square matrix, $e_1$ ...
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36 views

Matrix norm properties: inequalities

Is the fololwing relationship always true: $x^{\top} (A - \frac{1}{2} \|A\|_F \, I_3) x > 0$, knowing that matrix A is definite positive?
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70 views

Question about a proof in Rudin's book - annihilators

I'm reading the proof of the following theorem in Rudin's "Functional Analysis": Let $M$ be a closed subspace of a Banach space $X$. The Hahn-Banach theorem extends each $m^* \in M^*$ to a ...
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49 views

Max - min problem of a quotient of norms

For the $2\times2$ matrix $\begin{bmatrix}4&0\\-3&-5\end{bmatrix}$ Part 1 Find nonzero vectors $u$ and $w$ that maximize and minimize respectively the quotient $||Av|| / ||v||$. Part 2 ...
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55 views

Which is finer(larger) between the sequence spaces $l_{p}$ & $l_{p+1}$

Prove that, $l_{3}\subset l_{7}$ & $L_{9}[0,1]\subset L_{6}[0,1]$, where $l_{p}$ & $L_{p}[0,1]$ are of usual notation. Are the converses hold for both cases? Can these two results ...
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42 views

Verifying that the Sobolev space is a Banach Space

In PDE Evans 2nd edition (pages 262-263), I am trying to understand a proof of a theorem, which states: THEOREM 2 (Sobolev spaces as function spaces). For each $k=1,\ldots$ and $1\le p\le\infty$, ...
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26 views

matrix F-norm inequalities on matrix sum and product

As you know, we have the two following inequalities for sum and product (for Frobenius norm): $\|A+B\|_F\leq\|A\|_F+\|B\|_F$ and $\|AB\|_F\leq\|A\|_F\|B\|_F$. The question is, under which ...
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46 views

Reference for an inequality between $|x+y|^p$, $|x|^p$, $|y|^p$, and $|x-y|^p$

I am interested in an inequality for real numbers $x,y$ and $1<p<\infty$ that it should say something like $$ |x+y|^p -|x|^p - |y|^p \leq (1-2^{1-p})|x-y|^p. $$ Is this inequality (or ...
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31 views

How nuclear norm is convex whereas weighted nuclaer norm is not?

In (http://nuit-blanche.blogspot.in/2014/05/wnnm-weighted-nuclear-norm-minimization.html), it is stated that nuclear norm of a matrix $\mathbf{X}$, given as $||\mathbf{X}||_{*}=\sum_{i} ...
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50 views

Cauchy Schwartz inequality and absolute value

Here's the inequality: $$|\langle u,v\rangle|\le\|u\|\cdot\|v\| $$ Why on the LHS there's an absolute value? We know that $\langle u,v\rangle \ge 0$ Isn't it redundant?
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50 views

Product spaces $X = Y = \mathbb R$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. Let $d_{X \times Y} : X \times Y \rightarrow \mathbb R_+$ be given by $$d_{X \times Y}((x_1,y_1),(x_2,y_2)) := d_X(x_1, x_2) + d_Y(y_1, y_2)$$ How ...
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28 views

Dirichlet problem, solvability

Let $\Omega\subset \mathbb R^n$ and $f\in L^2(\Omega)$. We consider $\{u\in C^1(G):||u||_{1,2,G}<\infty \}$,where $||u||_{1,2,G}$ is the "sobolev-norm" with parameter $1,2$ on $G$. Then $H^1(G)$ ...
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51 views

Inequality$\Big|\sum_{j=1}^n a_{1j} x_j \Big|^2 \leq \sum_{j=1}^n |a_{1j}|^2 \sum_{j=1}^n |x_j|^2$

Let ${\bf A}$ be a $m \times n$ matrix with entries $a_{ij}$, and ${\bf x}$ be a $n \times 1$ vector with entries $x_{i}$. Then how can I show $$ \left\vert\,\sum_{j\ =\ 1}^{n} ...
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50 views

Norm for a set of vectors

Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$ I ...
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20 views

Product of $L_2$ norm of vectors

Is the $\sum \Vert b_k\Vert_2^2 \le\ge= \sum \Vert b_k\Vert_2^2 \Vert a_k\Vert_2^2$ ? where $b_k$ is a column vector and $a_k$ is a highly sparse row vector.
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37 views

Frobenius Norm to L2 norm conversion

Can anyone help me with a relation: I want to represent Frobenius norm of matrix $X$ in terms of l2 norm of constituent vectors $b$ and $a$ $\Vert X\Vert_F^2=\Vert ba \Vert_F^2$ where $b$ is a ...
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14 views

norm bounded Convolution in matrix space

There is a stable matrix $A$ with eigen values in unit circle,for discrete time system : $x(k+1)=Ax(k)+f(k)$ can we prove: $||\Sigma_{j=0,..,k} A^{k-j}f(j)||_2<= ||f(k)||_2/{(1-A_{max})} $ where ...
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27 views

Trace distance between “weighted” Hermitian matrices

The trace norm for a matrix $\mathbf{A}$ (also known as Shatten 1-norm) is defined as follows: $\|\mathbf{A}\|_1=\operatorname{trace}[\sqrt{\mathbf{A}^*\mathbf{A}}]$. It yields a useful distance ...
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29 views

Norm of an operator and eigenvalues

I have $K\colon L^2(0,T) \to L^2(0,T)$ a Hilbert-Schmidt integral operator (and so $K$ is linear, bounded, compact and self-adjoint) and I have obtained its eigenvalues and eigenvectors. From them, I ...
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43 views

Polynomials of best approximation

The question is about approximating the continuous function in an interval $[a, b]$. If we consider the linear space of all such functions endowed with the norm $$||f|| = \max_{x \in [a, b]}|f(x)|$$ ...
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30 views

Unit ball for a special norm

What does the unit sphere for the norm on $\mathbb{R}^2$, $\displaystyle N(x,y)\rightarrow\sup_{t\in\mathbb{R}}\frac{|x+ty|}{t^2+t+1}$, look like ? My approach was to consider $y=ax$ so as to get ...
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71 views

Minimizing the Frobenius Norm

I would like to minimize the following expression with respect to matrix X: $$\left \| A-BX \right \|_{F}$$ where A and B matrices are given and all the matrices have positive integer elements. Any ...
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24 views

Norm of linear functional in $W^{1,2}$

I want to find the norm of the following linear functional: $\phi:W^{1,2}_{[0,1]} \rightarrow\mathbb{R}$ $\phi(f)=\int^1_0t\cdot f(t)+ f^{'}(t) dt $ The norm is: ...
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63 views

Maximum eigenvalue of product of two matrices

Let $A$ and $B$ be two Hermitian matrices. I wanted to know if there is any relation between the maximum eigenvalue of $AB$ and that of $A$ and $B$. Is the following relation true? ...
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35 views

Greatest norm for the exponential of a polytopic matrix

im new to the forum and I have a question that Im working quite for a while now. I would like to prove that the greatest norm when considering the exponential ...
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68 views

Matrix norm induced by a vector norm.

All matrices are real. $A$ is a matrix of size $n \times k$ with $k < n$ and has independent columns. The function $v(x) = \|Ax\|_1$ is a norm. What is the matrix norm induced by $v$? Is it of ...
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51 views

What does it mean to write $|||x|||$ rather than $||x||$?

I am familiar with the notation $||x||$ meaning some norm of $x$. I have just come across the notation $|||x|||$ (in a text that also uses the former for norms). What is the difference?
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$\| AB\|_\square \leq 1$ implies $\| BA\|_\triangle \leq 1$

In this post norm denotes a matrix norm, i.e. it is sub-multiplicative. All matrices are real. $A$ is of size $n \times k$ with independent columns ($k \leq n$). $B$ is of size $k \times n$. Let $\| ...
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87 views

What is the general definition of the conjugate of a multiple-component number?

I know that the conjugate of a binomial is the negation of the second part. So the conjugate of (a + b) would be (a - b). I know that the conjugate of a complex number (a + bi), similarly, is (a - ...
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A question concerning the Schwartz space

Denote the Schwartz space by $\mathcal S(\mathbb{R})$. I want to show that $\forall n,k \in \mathbb{N} \cup \{0\}$, $\|\cdot\|^{(n,k)} : \mathcal S(\mathbb{R}) \rightarrow [0, \infty)$ defined by ...
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41 views

$\|u\| = \Big(\int |\nabla u|^p \Big)^\frac{1}{p}$ as a norm on $W_0^{1,p}(\Omega)$ where $\Omega$ is bounded

Let $\Omega$ be a bounded open set of $\mathbb{R}^N$, define $\|u\|_1 = \Big(\int |\nabla u|^p \Big)^\frac{1}{p}$ as a norm on $W_0^{1,p}(\Omega)$ for $2\leq p<\infty$, where $$\int |\nabla u|^p ...
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1answer
55 views

What is the matrix norm induced by weighted vector norm?

I denote vector norms with doulbe bars and matrix norms with triple bars. It is well known that the vector norm $L_2$ i.e. $\| x \|_2 = \sqrt{x^\top x}$ induces the matrix norm $||| \cdot |||_2$, ...
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1answer
55 views

derivative of a norm VS norm of a derivative

Consider a vector-valued function of the time, say $$v: \tau\in\mathbb{R}\to\mathbb{R}_N.$$ Suppose that for $\tau=t$, the function is equal to the zero vector, i.e. $$v(t)=0_N.$$ Denote as ...
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1answer
37 views

involution on $\mathbb F_p((t))$

Recently I learned here interesting facts about $\mathbb F_p((t))$, the field of the formal Laurent series with coefficients in $\mathbb F_p$. I looked up quite a few papers on local fields but I ...
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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 ...
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161 views

symbol for maximum number in an array or vector

As we know $\|x\|_{\infty}$ returns the maximum "absolute" value through the vector . I'm looking for a symbol to show the actual value that can be also negative: $ ?x?=\max\{x_1,...x_N\}$
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39 views

Upper bound for integral over boundary in terms of integral over interior

I've encountered quite some papers in which it is simply assumed that $\exists C>0 : \left(\displaystyle{\int\limits_{\Gamma}}((\nabla v)\cdot \hat{\bf{n}})^2d\Gamma\right)^{\dfrac{1}{2}}\leq ...
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41 views

Is this a norm for $C^1[0,1]$?

Let $C^1[0,1]$ be the space of differentiable functions in [0,1] in which their derivatives are continous in [0,1]. Define: $ |||f|||_1=||f||_\infty =max \left \{ |f'(x)|: 0\leq x \leq 1\right \} ...
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22 views

Is there a cost function for row equivalent matrices?

I am working on a minimization problem as follows: argmin$_x$ ||x-y||$_2$$^2$+$\lambda$||$\Psi$x||$_1$ where x and y are 2D or 3D complex arrays ||$\cdot$||$_1$ and ||$\cdot$||$_2$ are the L1 and L2 ...
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73 views

Show that $\lim_{p→∞} ||x||_p = ||x||_∞$

For any $x ∈ \mathbb{R}^n$ and $p ≥ 1$, define $$||x||_p = \left(\sum_{i=1}^n|x_i|^p\right)^{\frac1p},||x||_∞ = \underset{1≤i≤n}{\max}|x_i| $$ Show that $$ \underset{p→∞}{\lim} ||x||_p = ...
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36 views

Limits in matrix norm if convergence of integration is guaranteed

the answer of this question probably is very obvious, but I want to make sure this is correct. I have a function $F: \mathbb{R}\rightarrow \mathbb{R}^{n\times n}$ that is continuous and ...
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1answer
24 views

Norm inequality question

If $T$ is an invertible matrix, and for matrices $B$ and $C$, and we have that $∥TA−TB∥<ϵ$, can we say that A and B are close to each other in some sense? Intuitively the above shows that ...
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41 views

Generalized Form of Fano's Inequality

The Wikipedia article on Fano's Inequality presents a generalization as follows: Let $\mathbf{F}$ be a class of probability densities with a subclass of $r+1$ densities denoted $f_{\theta^{(i)}}$ ...
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38 views

Understanding an equality with open balls

In order to understand a proof I want to know why the following is true: Let X be a banach space and $x\in X$ with $x\in B(x_0,r)$ (open ball around $x_0$ with radius $r$), then $\frac{r}{2}x-x_0 \in ...
2
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1answer
24 views

“max” entry norm inequality?

Suppose $\max_{i,j}(A_{i,j}-B_{i,j})<t$. What can we say about $\max((TA)_{i,j}-(TB)_{i,j})$, where $T$ is a positive semi-definite matrix? I would like to say that ...
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89 views

How to prove a matrix norm inequality?

$P$ is a stochastic matrix i.e. square, non-negative, rows sum to one. Let $\Phi$ be a real matrix of size $n \times k$ with independent columns and $k < n$. Let $\Xi$ be the diagonal matrix with a ...