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

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How to compute 2-norm between two matrices of different sizes?

I have a matrix $A$ of size $n\times8$ and matrix B of size $m\times8$. I need to compute $2$-norm to measure similarity. It can be measured as: $d(A,B)= \frac{||A-B||_{2}}{||A||_{2} ||B||_{2}}$
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Proving a trivial bound on $L_2$ norm of the error in a sparse approximation of a vector

Trying to understand this supposedly 'trivial' bound from a paper: If $\theta_N$ denotes the vector $\theta$ with everything except $N$ largest coefficients set to $0$ then we have $$ || \theta - \...
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32 views

Is this a correct definition for $p$-adic norm?

The definition of $p$-adic norm in most textbooks and here is not easy for me to understand and especially to implement in practice, but here it is the way I reworded it: The norm of a $p$-adic ...
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3answers
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Dual norm intuition

The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it ...
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40 views

Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
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2answers
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Why is the Euclidian norm convex, if the square root function is concave?

I have some trouble figuring out if the Euclidean norm is convex. $\left\|{\boldsymbol {x}}\right\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}$ On one side I read that all norms are convex (...
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1answer
28 views

Operatornorm of $(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$

Determine the operatornorm of the mapping $I:(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$! Unfortunately I haven't many ideas for this task. I know that the definition of the ...
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3answers
36 views

Smallest possible value of the norm?

The vectors $ \vec{u_1} = \begin{bmatrix} 1 \\ 1 \\ 1\\ 1 \end{bmatrix} $ and $ \vec{u_2} = \begin{bmatrix} 1 \\ -1 \\ 1\\ -1 \end{bmatrix} $ are orthonormal in $ \mathbb{R}^4$....
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1answer
22 views

Is the condition number of unitary matrix always equal to 1?

I know that the 2-norm condition number $\kappa (\textbf U)={||\textbf U||_2}{||\textbf U^{-1}||_2}$ of a unitary matrix $\textbf U$ is always equal to 1. Is this true for all induced matrix norms, i....
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Generalizing norms: leaving out absolute homogeneity

Given a function $\rho:X\to\mathbb{R}$ on a vector space $X$ which satisfies the following properties: $\rho(x)=0$ if and only if $x=0$ $\rho(x+y)\leq\rho(x)+\rho(y)$ $\rho(-x)=\rho(x)$ for any $...
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3answers
361 views

L1-norm minimization

This is undoubtedly a trivial question but might as well ask: Why is the L1 norm minimization a heuristic for finding the sparsest vector? What I mean is that if the L1 norm sums the elements of a ...
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1answer
29 views

Why isn't $\ell^p$ locally convex for $0<p<1$?

I believe we have to distinguish the finite-dimensional from the infinite dimensional case. Regardless, if $0<p<1$, $\|x\|_p := (\sum |x_i|^p)^{\frac 1 p}$ is not a norm as it fails to satisfy ...
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1answer
25 views

Calculation of the squared Euclidean norm

Just a simple question. I'm reading an article and come across an equation that I cannot replicate as done in the original. \begin{align} &\|\mathbf{x} - \mathbf{\alpha} \|^2 - \|\mathbf{x} - \...
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9 views

Is this correct for an explicit represenation of the $\Vert f \Vert_{C^2(\mathbb{R}^2)}$ norm that uses multi-index notation?

I'm not familiar with multi-index notation so I'm not sure if I have this correct. Say we have (taken from here) $$ \Vert f \Vert_{C^2(\mathbb{R}^2)} = \sum_{j=0}^2 \sup_{x \in \mathbb{R}^2} |\nabla^j ...
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1answer
19 views

Distributive property on trace norm

I hope this is not a trivial question, basically, if we have trace norm of $A$ defined as $||A||_\star := \operatorname{trace}\left(\sqrt{A^*A}\right) = \sum\limits_{i=1}^{\min\{m,n\}} \sigma_i$, if $...
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54 views

Two inner products in one vector space.

Please, can you help me answer the some following questions? In theory functional analysis. At first, I want to consider finitely dimensional vector space V over field K(real or complex). Now, if it ...
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2answers
44 views

Inequality of Scalar Product involving derivative

I am stuck trying to reach an (in)equality... Let $\Omega \in \mathbb{R}$ and $f=f(t,x): \mathbb{R} \supset[0,T] \times \Omega \rightarrow \mathbb{R}$ be an element of $$\mathcal{W}:=L_{_t}^2(0,T,H_{...
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3answers
41 views

Show $d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$ is a metric on $C[0,1]$

I am surprised that this question hasn't been asked on here I need to show that $$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$ is a metric on $C[0,1]$ Proof: As usual, positive ...
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2answers
3k views

Proving the triangle inequality for the $l_2$ norm $\|x\|_2 = \sqrt{x_1^2+x_2^2+\cdots+x_n^2}$

I want to prove the triangle inequality for the l2-norm $\|x\|_2$: $$\|x\|_2 = \sqrt{x_1^2+x_2^2+\cdots+x_n^2}$$ $$\begin{align} \sqrt {\sum\limits_{i = 1}^n (x_i + y_i)^2 } &\leqslant \sqrt {\...
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1answer
18 views

Equivalence of weighted supremum norm

We define (for a measurable space $\Omega$, a bounded and measurable function $\phi$) a weighted supremum norm: $$|| \phi ||_\beta := \sup\limits_{x\in\Omega}\, \frac{|\phi(x)|}{1+\beta V(x)}$$ where ...
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1answer
27 views

Discriminant of real cyclotomic field

I know following theorem (and its proof): Let $K\subset L \subset M$ be number fields, $[L:K] = n, [M:L]=m$, and let $\{\alpha_1,\ldots,\alpha_n\}$ and $\{\beta_1,\ldots,\beta_m\}$ be bases for $L$...
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1answer
27 views

Norm of multiplication and multiplication of norms

It is well known that $\|u \cdot v \|_2 \le \|u \|_2 \cdot \| v \|_2 $ for all $u, v \in \mathbb{R}^d$. Is the following true for all $p \in \mathbb{R}$: $$\|u \cdot v \|_p \le \|u \|_p \cdot \|...
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13 views

Unknown normalization

I am trying to do BPP imaging and have been trying to replicate the algorithm in "Photoplethysmographic imaging of high spatial resolution" however I've ran into a problem. They say they normalize ...
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21 views

Is following a norm or absolute value of a vector?

I'm reading a paper regarding power minimization and came across with following equation: $g_{i,j}=|h_{i,j}|^2/d^\alpha$ Where $h_{i,j}$ is a complex vector of dimension $N$. I don't know and it ...
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1answer
28 views

For what value $p$ is $l_p$ not a norm or a metric? [closed]

Can someone please remind me for which values of $p \in [0, \infty)$ is the little $l_p$ norm or $l_p$ metric not a norm or a metric I vaguely remember that $l_0$ norm is a not a norm. Could someone ...
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1answer
39 views

inequality between operator norm and infinity norm

For a matrix $A$, is there any relation between operator norm (https://en.wikipedia.org/wiki/Matrix_norm#Induced_norm) and infinity norm (defined as the maximum of the absolute value of all the ...
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When can I say that $\overline{A} \subset B$ if I know that $A \subset B$?

my question is as stated in the title: When can I say that $\overline{A} \subset B$ if $A \subset B$? Here $A,B$ are normed spaces and the closure of A is taken with respect to the norm of B. Can I ...
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1answer
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Bounding sum of norms by the sum of sqaure of norms [closed]

How can you bound sum of norms (e.g. sum of norms of vectors) by sum of square of the same norms? Please advise. Thanks in advance.
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Which of the followings are norms on $\mathbb R^2 ?$

$||x||_1=|x_1|+|x_2|$, where $ x=(x_1,x_2)\in \mathbb R^2$ $||x||_2=\displaystyle\sqrt {x_1^2+x_2^2}$, $ x=(x_1,x_2)\in \mathbb R^2$ $ ||x||_{max}=max\{||x||_1,||x||_2\}$,$ x=(x_1,x_2)\in \mathbb R^2$...
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closeness of matrices

I'm really lost in math and would really appreciate any help with the following problem. Denote as $S_{+}(p)$ the set of all positively defined symmetric real-valued matrices of size $p \times p$. ...
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1answer
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A fast randomized algorithm for the approximation of matrices

I am reading a paper which title is the same as the title of this question. Demonstration of Lemma 3.13 in the paper says that $ P\left ( \frac{\left \|Ax^{(j)} \right \|}{\left \| x^{(j)} \right \|...
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Induced matrix 2-norm - restricted direction

I have a problem, let: $M = \begin{bmatrix} m_{11} & m_{12} & m_{13} & m_{14} \\ m_{21} & m_{22} & m_{23} & m_{24}\end{bmatrix}$ and $X$ is general matrix of size [4x2]: ...
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Optimization with L_infinity norm regularization

I'm trying to solve an optimization problem of the form $$\text{minimize } \; f(x) + \|x\|_\infty$$ where $x$ ranges over all of $\mathbb{R}^n$ and $f:\mathbb{R}^n \to \mathbb{R}$ is a nice, smooth, ...
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1answer
543 views

Matrix inner product, and operator and trace norm inequality

I have trouble proving the following inequality. Let a matrix $A \in \mathbb{R}^{M \times N}$, and $\sigma_i(A)$ be the i-largest singular value of A. Define the operator norm and the trace norm as ...
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1answer
22 views

Simple norm inequality

Trying to follow the comments to this question I am struggling very much to understand how to simplify $\|Ax\|_2=\sup_{\|x\|_2=1}\sqrt{\sum_i(\sum_ja_{ij}x_j)^2}$ to arrive at an $x$-free bound. Can ...
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2answers
118 views

A lower bound for the ratio of $2$- and $\infty$-norms within a linear subspace

Let $M$ be a $k$-dimensional linear subspace of $\mathbb{R}^n$. Define its "distortion" (with respect to $2$- and $\infty$- norms) as $$d(M)=\sup_{x\in M\setminus \{0\}}\frac{\|x\|_2}{\|x\|_\infty} = ...
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28 views

Difference between norm and distance. [duplicate]

I was wondering the difference between norm and distance. My teacher told me that a norm always induce a distance, but that the reciprocal is not true. So, let $(E,\|\cdot \|)$ a normed space. I agree ...
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47 views

contraction mapping proof

I was reading a paper, and there is a proof I don't understand. How did they get from eq(6.4) to eq(6.5) using the norm they defined? Any help would be appreciated. Thanks in advance! Here is the ...
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choose between L1 and L2 normalization in logistic regression regularization

Wondering what are the pros and cons comparing to L1 and L2 normalization in logistic regression regularization part, For example, in below formula, it is use L2 normalization (in squared form of ...
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1answer
19 views

Spectral radius of block-skew-hermitian matrix equals norm of block

$$\rho\left(\left[\begin{matrix}0 & A \\ -A^{\dagger} & 0\end{matrix}\right]\right)=\|A\|$$ where $\rho(\cdot)$ is the spectral radius, $\|\cdot\|$ is the induced 2-norm. Question: I am ...
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1answer
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Prove a series finite a.e by proving that its $L_{1,\infty}$ norm is finite.

According to a article, we can show that series $\sum_{i=1}^\infty 2^i\left(\mathbf{1}_{A_i}\right)(x)$ is finite almost surely by proving that its $L_{1,\infty}$ norm is finite. Can you explain me ...
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1answer
908 views

Poincaré inequality in unbounded domain

Help me please, how can I show that Poincaré inequality doesn't hold in an unbounded domain? Thanks a lot! If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...
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What are the different ways in which I can find Lipschitz constant for:

What are the different ways in which I can find Lipschitz constant for $$|| \bigtriangledown(X)||_F^2$$ where $$|| \bigtriangledown(X)||_F^2 = \sum_{i,j}|| \bigtriangledown(X)_{i,j})||_F^2$$ and $\...
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1answer
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How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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1answer
46 views

Operator norm under shrinkage

If I have a $n$-dim matrix $A=\{a_{ij}\}$, and I multiply each elements by a factor $w_{ij}$ in $[0,1]$, and get a new matrix $A_w=\{a_{ij}w_{ij}\}$. Do I have $$||A||\ge \lVert A_w\rVert$$ where the ...
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2answers
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If the entries of a positive semidefinite matrix shrink individually, will the operator norm always decrease?

Given a positive semidefinite matrix $P$, if we scale down its entries individually, will its operator norm always decrease? Put it another way: Suppose $P\in M_n(\mathbb R)$ is positive ...
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1answer
29 views

Proof $\sigma_{\min}(A\Delta)\geq\sigma_{\min}(A)\sigma_{\min}(\Delta)$, $\sigma$ is a singular value

Let $A$ and $\Delta$ be square matrices. The definition of smallest singular value of a matrix $A$. (in title, $\sigma_{\min}$): The matrix norm is the 2-induced norm. The propertie: I don't ...
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Prove that $\bigg\| \begin{bmatrix} X \\ A\end{bmatrix} \bigg\|_2 \leq \sigma \iff X^* X + A^* A \preceq \sigma^2 I$

Prove that $$\bigg\| \begin{bmatrix} X \\ A\end{bmatrix} \bigg\|_2 \leq \sigma \iff X^* X + A^* A \preceq \sigma^2 I$$ Here, * denotes the conjugate transpose. This norm is the $2$-induced ...
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0answers
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Norms (eigenvalues) of sums of orthogonal matrices

Let $T_1, \ldots, T_n$ be a set of real-valued symmetric matrices satisfying $Tr(T_j T_k) = 0$ for all $j\neq k$. Consider the norm $\|T\|_{\infty} = \max_{\|M\|_1 \leq 1} \operatorname{Tr}\left[M^T ...
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1answer
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Finding the global minimum

Let $f~:~\Bbb R^2\to \Bbb R$ be defined as: $$f(x)=\left\|\begin{bmatrix}2&1\\3&1\\4&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix} - \begin{bmatrix}2\\1\\7\end{bmatrix}\right\|_2^2$$ ...