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

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$L^p$ norm of multivariate standard normal random variable

Given $X_i\sim \mathcal{N}(0,1)$ what is the behaviour of $$ ||X||_{l^p}=(\sum_{i=1}^n|X_i|^p )^{1/p}$$ as $n\rightarrow \infty$? For $p=2$ results about $\chi$-distribution tell us that ...
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3answers
130 views

Proving $|x|$ is a norm in $\mathbb {R}^n$

I need some direction on how to start on showing that $| x+y|\leq|x|+|y|$ in $\mathbb R^n$. Note that $$ |x|=\left(\sum\limits_{j=0}^n x_i^2\right)^{1/2} $$ Thank you, Klara
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1answer
1k views

Trace Norm properties

Let $||A||_1=\operatorname{trace}(\sqrt{A^* A})$. I already proved that for arbitrary unitary matrices $U$ and $V$, $||UAV^*||_1=||A||_1$ and $||A||_1=\sigma_1+\dots+\sigma_k$. Now I would like to ...
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1answer
250 views

The openness of the set of positive definite square matrices

Let $\mathbb{R}^{n\times n}$ be the vector space of square matrices with real entries. For each $A\in \mathbb{R}^{n\times n}$ we consider the norms given by: $$ \displaystyle\|A\|_1=\max_{1\leq j\leq ...
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92 views

Help with an operator norm

Let $T\in \ell_\infty(\mathbb{Z,\mathbb{C}})^*$ such that: $T(1_{\ell_\infty})=1$ where $1_{\ell_\infty}$ denotes the constant function $1$; $T(u)\geq 0$ whenever $u$ is real positive. How to ...
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2answers
403 views

How to prove positive definiteness?

$B_{(n+1)(n+1)}$ = $ \begin{bmatrix} A & u \\ u^T & 1 \\ \end{bmatrix}$ is given, and $A$ is a positive definite matrix where its Cholesky factorization is given ...
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1answer
481 views

Multiplicative norm on $\mathbb{R}[X]$.

How to prove that : there is no function $N\colon \mathbb{R}[X] \rightarrow \mathbb{R}$, such that : $N$ is a norm of $\mathbb{R}$-vector space and $N(PQ)=N(P)N(Q)$ for all $P,Q \in \mathbb{R}[X]$. ...
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170 views

Is norm non-decreasing in each variable?

Let me try again. Suppose $\|\cdot\|$ is a norm in $\mathbb{R}^n$ and let $$f(x_1,...,x_n)=\|(x_1,...,x_n)\|$$ where $x_i\geq 0, \forall i$. I want to prove or disprove that $f$ is an nondecreasing ...
6
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0answers
74 views

Proof that $\|(a,b)\| \leq \|(c,d)\|$ if $0 \leq a \leq c$ and $0 \leq b \leq d$ [duplicate]

Possible Duplicate: Is norm non-decreasing in each variable? Let $\| \cdot \|$ be any norm on $\mathbb{R}^{2}$. Let $0 \leq a \leq c$ and $0 \leq b \leq d$. Show that $\|(a,b)\| \leq ...
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1answer
38 views

Prove that $\|{e^{At}x_o}\| \geq e^{-\lambda t}\|{x_o}\|$

Prove that $\|{e^{At}x_o}\| \geq e^{-\lambda t}\|{x_o}\|$, for some $\lambda \gt 0$, A is $n\times n$ matrix and $x_o$ is a $n \times 1$ vector.
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2k views

Relations between p norms

The p-norm is given by $||x||_{p} = (\sum_{n=1}^{\infty }|x_{n}|^{p})^{1/p}$. For $0 < p < q$, it can be shown that $||x||_{p} \geq ||x||_{q}$ (1, 2). It sppears that in $R^{n}$ a number of ...
3
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0answers
211 views

Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation

Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$. How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
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1answer
511 views

prove that a function is an inner product

I would appreciate some assistance in answering the following problems. We are moving so quickly through our advanced linear algebra material, I can't wrap my head around the key concepts. Thank ...
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1answer
95 views

A metric on $\mathbb{R}^n$ such that $d(\lambda x, \lambda y)=|\lambda| d(x,y)$ which is not induced by a norm

Let $V=\mathbb{R}^n$. Let $d:V \times V\rightarrow \mathbb{R}$ a metric on $\mathbb{R}^n$. Assume that for any $x,y\in V$ and $\lambda \in \mathbb{R}$, we have $d(\lambda x, \lambda y) = ...
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0answers
65 views

Showing inner product comes from a norm defined using Polarization Identity [duplicate]

Possible Duplicate: Norms Induced by Inner Products and the Parallelogram Law Trying to prove if an norm satisfies the Parallelogram Law then a norm arises from an inner product. Let ...
4
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1answer
650 views

Norm of integral operator in $L^1$

What is the norm of the operator $$ T\colon L^1[0,1] \to L^1[0,1]: f\mapsto \left(t\mapsto \int_0^t f(s)ds\right) $$ ?
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98 views

Discrete fourier transform and Frobenius norm

I have thousands of data files which are essential DFT data of plots such as this: And a DFT of the plot with a "Hard" threshold of 0.9 gives me: This DFT is just the left top corner of the DFT ...
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0answers
127 views

Upper bound for L1-L2 optimization problem

I am interested in the following convex optimization problem: \begin{align*} \max & ||x||_1 \\ \text{s.t.} & ||x-a||_1 \le K \\ & ||b\circ x||_2 \le 1\\ & x \in R^n \end{align*} where ...
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1k views

Why is this true: $\|x\|_1 \le \sqrt n \cdot \|x\|_2$?

Stuck, help, please, I am really new to this. I opened the 2-norm and multiplied by $n$, then I am thinking to square both sides. The problem is that I do not know how to open $(x_1 + x_2 + ... + ...
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2answers
122 views

My “wrong” comparison between $\ell^2$ and $\ell^1$

For sure, $\ell^2$ is larger than $\ell^1$, because for $|x|<1$, $|x|^2<|x|,$ that is, $||x||_2\leq||x||_1.$ But using Cauchy-Schwartz inequality, I get a "wrong" comparison: ...
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1answer
190 views

Equivalence of Schatten and spectral norms

I'd like some help showing the equivalence of these two norms when $p = \log n$. Recall the $p$-th Schatten norm of a linear operator $A$ acting on $\mathbb{R}^{n}$. In the particular case of $p = ...
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1answer
243 views

Do Lipschitz-continuous funcions have weak derivatives on bounded open sets?

Let $\Omega\in\mathbb{R}^n$ be open and bounded. I'm wondering if a function $f\in C^{0,1}(\Omega)$ (a Lipschitz-continuous one) is also an element of $W^{1,2}(\Omega)$ (that is the space of weakly ...
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1answer
152 views

Different norms for space $C[0,1]$

For the space of all continuous functions we can have the sup norm: $|f|=\sup|f|$ I have also seen the following norm: $|f|=\sup|f(x)|/|x|$ I don't know what this norm is called and therefore can't ...
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1answer
177 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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425 views

Example of a sequence that converges to two different limits with respect to two complete norms

I've wondered about the following question : Is there an (explicit?) example of a vector space $X$, two complete norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on $X$, and a sequence $(x_n) \subseteq X$ such ...
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5answers
167 views

Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
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88 views

Norm in $L^2$ space

I have a brief question regarding the norm of the $L^2$ space defined on an interval $[a,b]$. On various websites I have seen this defined as: $$\|f(x)\| = \int_{a}^{b} f(x)^2 dx$$ However, ...
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0answers
54 views

Why is $z^T x \le \|x\| \| z \|_*$ for dual norm in $R^n$?

This is probably very obvious, I was looking at this at this link. 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 ...
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1answer
50 views

Does the norm have a specific name?

Does the norm $$\|f\|=\sup\limits_{t\in[0,T]}\int\limits^t_0|f(\tau)|\ d\tau$$ have a specific name?
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1answer
33 views

How to implement fuzzy minimum function via fuzzy maximum

I know that I can represent fuzzy max via power function(i need it in neural network) i.e. def max(p:Double)(a:Double,b:Double) = pow(pow(a,p) + pow(b,p) , 1/p) // assumption a >=0 and b >=0 ...
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1answer
79 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 ...
2
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1answer
223 views

$C^1 [0,1]$ with different norm

If the space $C^1 [0,1]$ is equiped with norm $\Vert \cdot\Vert_1$,where $$ \Vert f\Vert_1=|f(0)|+\Vert f'\Vert _{C}=|f(0)|+\sup_{t\in [0,1]}|f'(t)| $$ for any $f\in C^1 [0,1]$, is this space Banach? ...
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1answer
1k 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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2answers
264 views

Norm and continuous functions

Having the definition: A function $f:\mathbb{R^n}\rightarrow \mathbb{R^n}$ is proper if $\|f(x)\|$ tends to $\infty$ when $\|x\|$ tends to $\infty$. I have to show : a)If $f:\mathbb{R^n}\rightarrow ...
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1answer
172 views

Solution for a Frobenius norm inequality

Am trying to find a real scalar $\gamma$ such that for a given pair of real rectangular matrices $X,Y$ the following holds: $\frac{||Y||_{F}^{2}}{5} \leq ||\gamma X||_{F}^{2}\leq ||Y||_{F}^{2}$ ...
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1answer
72 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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1answer
5k views

Infinity matrix norm example

I have a brief question regarding the infinity matrix norm. The subordinate matrix infinity norm is defined as: $$\|A\|_{\infty} =\max_{1 \leq i \leq n}\sum_{j=1}^{n}|a_{ij}|.$$ This is derived ...
2
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1answer
202 views

inequality between norms of vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. It is known that $\|a\|_{\infty}\leq \|a\|_2\leq \sqrt n\|a\|_{\infty}$. Let $k<<n$. For which kind of vectors the following would be true: $$ ...
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1answer
454 views

Proving norm inequality

There is a brief proof in my textbook that I have one question about. We are supposed to prove that $||x||_{1} \leq n||x||_{\infty}$ for $x \in \mathbb{R}^n$ The book writes the following: ...
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135 views

Inequality involving norm and inner product

I am stuck proving this trivial inequality: on a real inner product space, $(||x||+||y||)\frac{\langle x,y\rangle}{||x|| \cdot ||y||}\leq||x+y||$ I have tried to square both sides and use the Cauchy ...
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1answer
307 views

Norm of a matrix

Suppose $A$ is an $n\times n$ matrix such that $A+A^H-\delta I_n$ is positive-semidefinite, for some $\delta>0$, then can we show a bound on the norm of $A^{-1}$ ? Can we show that this the norm of ...
3
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1answer
128 views

Bizarre formula for arc length

I'm reading a book on computer science/math and I found this formula for arc lengths that I've not been able to decipher: $$\left|\int_p^q\left\| {df(x)\over dx} \right\| dx\right|$$ where $\lVert ...
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1answer
100 views

Equivalence of Norms Question

I have a matrix where the $L_1$ norm on an row is equal to zero. My question is what can I say about the $L_2$ norm on any row of that matrix? Numerically with the example I have, computing the $L_2$ ...
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2answers
37 views

Where is the square in the Least square regression method?

I'm having a serious doubt in the least square regression problem. I guess its got to do with the notation of norm. Is the least square formulation $||b - \mathbf{A}x||^2$ or is it $||b - ...
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1answer
294 views

Symmetrically make this matrix orthogonal, but don't you dare use the Frobenius norm…

I have read many of the questions already here in regards to the Frobenius norm, but they do not help me too much. My question is, why is the Frobenius norm not considered a 'proper' norm? In a ...
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2answers
242 views

equivalence of two definitions of norm equivalence: “$|\cdot|_1=|\cdot|_2^\alpha$” vs. “being a Cauchy sequence is the same for both norms”

Let $|\cdot|_1$ and $|\cdot|_2$ be two norms on a field $\mathbb F$. We call the two norms equivalent if every Cauchy-sequence with respect to $|\cdot|_1$ is also a Cauchy-sequence with respect to ...
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0answers
178 views

Bound on euclidean norm

Is it possible to find a suitable lower bound on $$\left|\left(tM+\sum_{k=2}^\infty\frac{(tM)^k}{k!}\right)\cdot b\right|$$ for $M$ as $n \times n$ matrix, $b$ as $1 \times n$ vector and for all $t$ ...
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100 views

How to prove an inequality for a special structure of strictly triangular matrix

The problem I cause is attached below. I am trying to prove the inequality. By using small $M$, I found that the terms on the left side of the inequality are part of the terms of the expansion on ...
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2answers
74 views

$l_1$ norm projection with regularization term

I recently encountered an optimization problem and looking for some technical paper for the same.The problem is give as below, $\min f(x)+\lambda*r(x) $ $\ s.t \ x \geq 0, ||x||_1 = 1$. where $x$ ...
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1answer
172 views

Representing with Hilbert Schmidt Norm

Am trying to see, if the following Trace function can be expressed using a Hilbert Schmidt Norm: $\operatorname{Tr}(X^TAX)$. Here, $X$ is a matrix whose entries take values that are finite and reals ...