# Tagged Questions

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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### 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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### 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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### 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 (...
1answer
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### 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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### 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 $\mathbb{R}^4$....
1answer
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### 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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### 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]: ...
1answer
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### 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 ...
2answers
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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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### 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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### 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 ...
1answer
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### 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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### 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 ...
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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### What are all the Banach algebras where $\|a\|\|a^{-1}\|=1$? [closed]

Is there a characterization for all Banach algebras such that $\|a\|\|a^{-1}\|=1$ whenever $a$ is invertible?
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### A confusion about the definition of the “trace” norm

Given a $n \times m$ real matrix $A$ of rank $r$ one can define its SVD as $A = UD V^T$ with $D$ being a $r \times r$ diagonal matrix and $U^TU = V^TV= I$. Here clearly the diagonal entries of $D$ are ...