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

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Matrix expansion does not decrease norms

Given a block matrix $A = \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}\\ {{A_{12}}}&{{A_{22}}} \end{array}} \right]$, where $A \in {R^{N \times N}}$, it is true that the euclidean ...
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80 views

Inner product spaces of smooth functions

In the space $C^1([0,1])$ where each $f$ is an element of the space and $$||f||= \left(\int_0^1\left(|f|^2+|f'|^2\right)dx \right)^{1/2}$$ How can it be shown that $||\cdot||$ is a norm of the space?
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How to generate a N*D random matrix with columns of unit length?

Is it possible to generate a N*D random matrix with columns of unit length? If not, I also think it is possible of generating a N*D random matrix and, after that, normalizing it in order to have ...
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463 views

norm of sobolev space $H^{1/2}$

Let $\Omega\subset\mathbb R^d$ a Lipschitz domain and $\Gamma:=\partial\Omega$. For $u\in C^{\infty}(\Gamma)$ we define $$||u||_{H^{1/2}(\Gamma)} = \inf_{\substack{v\in H^1(\Omega) \\ v|_\Gamma =u}} ...
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62 views

Simple norm equivalence

Please I need help with this problem. Let $V$ be the vector space consisting of functions which are continuous over the interval $[0,1]$, take the value $0$ at the points $0$ and $1$ and are once ...
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70 views

What is $L^1$ norm for $n$-fold functional space of integrable function?

Consider a space of $n$-fold real-valued integrable functions, $X \doteq [L^1([0,T])]^n$. Some use an alternative notation like $X \doteq L^1([0,T]; \mathbb{R}^n)$. Does $f\in X$ mean ...
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37 views

Matrixnorm, “max” function

According to my lecture notes, the following equality holds: $ \frac{1}{\sqrt{n}}$ $\max_{x \neq 0}$ $\frac{\|Ax\|_{2}}{\frac{1}{\sqrt{n}}\|x\|_{2}}$ = $\max_{x \neq 0}$ ...
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364 views

Submultiplicative Matrix Norm

Given $I$ is the $n\times n$ identity matrix, $F$ an $n\times n$ matrix with $\|F\|_{M} < 1$ for some submultiplicative norm $\|.\|_M$, by which formula is $(I+F)^{-1}$ calculable? And is it ...
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102 views

Sobolev spaces doubt

Can somebody help me with this doubt? Let $\Omega$ an open set and $A$ be any finite subset of points of $\Omega.$ Is it true the following inequality? $\vert v(a) \vert \leq C \| v \|_p ...
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57 views

Norm to evaluate precision

I have two vectors x and y with their respective n coordinates/components and this norm between them: $$ norm = \sqrt{\frac{\displaystyle\sum_{i=1}^{n} (x_{i}-y_{i})^2}{\displaystyle\sum_{i=1}^{n} ...
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Norm of normal Operator A

I just found the following equality $||A||=sup_{\lambda\in\sigma(A)} |\lambda|$ My question: What exactly means $\sigma(A)$ and why this is true ? I always thouht the only way to get the ...
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140 views

Hilbert norm and Euclidean distance

For real matrix $X$ where $d_{i,j}^2(X)$ indicates the euclidean distance squared between the rows $i,j$ of $X$, if $d_{i,j}^2(X)=||f(X_i.)-f(X_j.)||_H$ then what would the function $f(.)$ be? Is ...
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171 views

Inequality for singular value for differences of matrices (upper bound)

Does anybody know the inequality of singular value for differences of matrices, i.e. $\sigma_{max}\left(\begin{array}{c} A-B\end{array}\right)\leq??$ in term of $\sigma_{max}\left(\begin{array}{c} ...
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131 views

How to find if the vector is stable?

Let's say I have the following equation $Ax=b$ My question is - how can i find a vector $b$ around which the above equation is not stable? I have $$A= \begin{bmatrix} 1 & 0.999 \\ ...
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89 views

Bounding L2 distance with mean and standard deviation

Let $\mathbf{x}=[x_i]_{i=1}^d, \mathbf{y}=[y_i]_{i=1}^d$ be two vectors in $R^d$. Is it possible to find a lower bound $l\leq \|x-y\|$ and an upper bound $u\geq\|x-y\|$ as a function of ...
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299 views

Determining a bound on the relative error of a linear system

Suppose that the linear system $A\mathbf{x}=\mathbf{b}$ is perturbed so that $(A+\delta A)\mathbf{x}=\mathbf{b}$. We can calculate the relative error $\frac{\|\mathbf{x-\bar{x}\|}}{\|\mathbf{x}\|}$ ...
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65 views

Why is $z^T x \leq \|x\| \| z \|_*$ for dual norm in $\mathbb{R}^n$?

This is probably very obvious, I was looking at this. It looks so much like a Cauchy-Schwarz though, and I would say it is very obvious from the definition if it wasn't for the condition that $\|x\| ...
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92 views

inequality between entries of the vector and $l_2$ norm of the vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. I am wondering for which vectors the following would be true: $$ \|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n. $$ Here $c>0$ is ...
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2k views

Matrix norms and spectral radius

Recently I started exploring convergence of some iterative methods and spotted the equivalent of the spectral radius and a matrix norm. For instance, ...
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82 views

Scalar multiplication and Frobenius norm

Was wondering on what would be the real number (scalar) $\gamma$ that needs to be multiplied with each entry in a real rectangular matrix $X_{m\times n}$ such that the Frobenius norm of $X$ equals a ...
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112 views

Is there a linear function that is *not* continuous between two normed vector space?

The textbook says that this function has to be continuous at least in the origin for it to be continuous everywhere. But how is it possible that a function is already linear but somehow not ...
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91 views

Reading a Laplacian Matrix and its labeled graph?

How can the following labeled graph be extracted from the Laplacian Matrix below and viceversa? I had a look at this great conversation but it is already too advanced for me.
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241 views

How to find/parameterize vector perpendicular to circle of constant $\ell_p$ norm

This should be very easy, but I can't get my head around it: given $1\leq p < \infty$, and a point x with $\|x\|_p = 1$, how do I get a (or the unit, or any) vector which is perpendicular to the ...
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48 views

A simple estimation

Let's say $X$ is a normed linear space, and $X^*$ is its dual space. One can define the norm in $X$ in such a way $$ \|x\| = \sup\{|\phi(x)|:\phi \in X^*,\|\phi\|\le 1\}. $$ The direction $$ ...
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Normed space and Subspace

Let $X$ be a vector space with norm and $Y$ subspace with non empty interior ($Y^\circ\neq\varnothing$), then $X=Y$. I'm trying to use the theorem that says: if $X,Y$ normed spaces and $T:X\to Y$ ...
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>>A. $X$ banch. >>B.M banach. >>C. $X/M$ banach.

I want to prove that any two will imply the other: A. $X$ banch. B.M banach. C. $X/M$ banach. I have proved A+B imply C by taking M closed in X. I am not understanding ...
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Norm and Cauchy sequence.

If someone could please show me how to show that if $x_n$ is a Cauchy sequence then $x_n \over ||x_n||$ is a Cauchy sequence as well? Thanks, I hope I've been clear enough.
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25 views

Bounded Operator and p-norm (more difficult than it seems).

Let $\mathbb{R}^k$ and $\mathbb{R}$ be real vector spaces (with the usual operations of addition and scalar multiplication in each one of them) with the norm $\|\mathbf{x}\|_p$ for the space ...
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29 views

Norm of Outer Product

Let $x \in \mathbb{R}^N$ and $ y\in \mathbb{R}^M$. Show that $\|xy^T\|_{\infty}=\|x\|_{\infty}\>\|y\|_1$ I've been able to show the following: $\|xy^T\|_{\infty}= \|xIy^T\|_{\infty} \le ...
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L0 norm, L1 norm and L2 norm

For vector $\boldsymbol{x},\boldsymbol{y} \in \mathcal{R}^{n}$, if \begin{equation} \| \boldsymbol{x} \|_0 = \| \boldsymbol{y} \|_0 \end{equation} What relationship will $\| \boldsymbol{x} \|_1$ and ...
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Norm of Integral Operator on $E=\{u \in C[0,1]\ ,\ u(0) = 0\}$

There are similar question but the characterization of the space $E$ that I have gives me problem in computing the actual norm. Let $E=\{u \in C[0,1]\ ,\ u(0) = 0\}$ with the usual $\parallel ...
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Chebyshev's Inequality and the length of a random vector

Suppose that we take $n$ iid random variables $X_1,\dotsc, X_n\sim \operatorname{Unif}[-1, 1] $ and define $Y_n=\lVert (X_1,\dotsc, X_n) ^\intercal \rVert $. By what means can we employ ...
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when does the equality hold for the matrix norm triangle inequality and product inequality

So here is the problem: When does the equality hold for the following two famous "matrix" norm inequalities: $\|A+B\|\leq \|A\|+\|B\|$ $\|CD\|\leq\|C\|\|D\|$ For any norm you prefer. But I'm ...
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solver for non-convex matrix optimization with convex constraints

So here is the problem: $\max_{D} ~~ \|A+BD\|$ subject to $\|D\|<1$ (any norm you like) where matrices A and B are given. The cost function is evidently convex as well as the constraint, but ...
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21 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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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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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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18 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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17 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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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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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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2answers
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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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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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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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21 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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19 views

matrix two norm derivative with respect to X

What would be the result of the following derivative in terms of X? $\frac{d \|X\|_2}{d~ X}=?$
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Composition of two functions in normed spaces

Let $\Omega_1, \Omega_2 \subset \mathbb{R^n}$ be bounded. The mapping $ F: \Omega_1 \rightarrow \Omega_2 $ shall be bijective, continuously differentiable and such that $||DF(x)||$ and ...
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Norm in $C(X,\Bbb{R})$

Let $X\subset\Bbb{R}$ a compact set and $f\in C(X,\Bbb{R})$. Define $$\|f\|_{\infty}=\sup A_f$$ with $A_f=\{|f(x)|\in \Bbb{R};x\in X\}$. Then $\|f\|_{\infty}=|f(x_0)|$, for some $x_0 \in X$, since ...
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8 views

Calculate 1-norm of a vector using another matrix or vector

Let's say I have a vector a. I would like to construct a matrix or vector b such that if I multiply ...
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15 views

the number of non-zero integral ideals of norm m in a ring of integers [closed]

How to prove that the number of non-zero integral ideals of norm m in a ring of integers of a number field with degree n is less than or equal to the number of n-dim vectors of n positive integer ...