# 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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### 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 (...
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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 $\mathbb{R}^4$....
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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]: ...
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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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### 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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### 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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Let $M$ be a $k$-dimensional linear subspace of $\mathbb{R}^n$. Define its "distortion" (with respect to $2$- and $\infty$- norms) as $$d(M)=\sup_{x\in M\setminus \{0\}}\frac{\|x\|_2}{\|x\|_\infty} = ... 0answers 28 views ### Difference between norm and distance. [duplicate] I was wondering the difference between norm and distance. My teacher told me that a norm always induce a distance, but that the reciprocal is not true. So, let (E,\|\cdot \|) a normed space. I agree ... 0answers 47 views ### contraction mapping proof I was reading a paper, and there is a proof I don't understand. How did they get from eq(6.4) to eq(6.5) using the norm they defined? Any help would be appreciated. Thanks in advance! Here is the ... 0answers 26 views ### choose between L1 and L2 normalization in logistic regression regularization Wondering what are the pros and cons comparing to L1 and L2 normalization in logistic regression regularization part, For example, in below formula, it is use L2 normalization (in squared form of ... 1answer 19 views ### Spectral radius of block-skew-hermitian matrix equals norm of block$$\rho\left(\left[\begin{matrix}0 & A \\ -A^{\dagger} & 0\end{matrix}\right]\right)=\|A\|$$where \rho(\cdot) is the spectral radius, \|\cdot\| is the induced 2-norm. Question: I am ... 1answer 16 views ### Prove a series finite a.e by proving that its L_{1,\infty} norm is finite. According to a article, we can show that series \sum_{i=1}^\infty 2^i\left(\mathbf{1}_{A_i}\right)(x) is finite almost surely by proving that its L_{1,\infty} norm is finite. Can you explain me ... 1answer 908 views ### Poincaré inequality in unbounded domain Help me please, how can I show that Poincaré inequality doesn't hold in an unbounded domain? Thanks a lot! If \Omega is a bounded domain and u \in H_{0}^{1}(\Omega) the following inequality ... 0answers 11 views ### What are the different ways in which I can find Lipschitz constant for: What are the different ways in which I can find Lipschitz constant for$$|| \bigtriangledown(X)||_F^2$$where$$|| \bigtriangledown(X)||_F^2 = \sum_{i,j}|| \bigtriangledown(X)_{i,j})||_F^2$$and \... 1answer 1k views ### How to describe minimization of L1 norm error using linear programming? Given a set of n pair points (x_1, y_1), ..., (x_n, y_n) in the plane, I need to find a line ax + by = c that fits the points of the L1 norm error points as closely as possible. I need a linear ... 1answer 46 views ### Operator norm under shrinkage If I have a n-dim matrix A=\{a_{ij}\}, and I multiply each elements by a factor w_{ij} in [0,1], and get a new matrix A_w=\{a_{ij}w_{ij}\}. Do I have$$||A||\ge \lVert A_w\rVert$$where the ... 2answers 73 views ### If the entries of a positive semidefinite matrix shrink individually, will the operator norm always decrease? Given a positive semidefinite matrix P, if we scale down its entries individually, will its operator norm always decrease? Put it another way: Suppose P\in M_n(\mathbb R) is positive ... 1answer 29 views ### Proof \sigma_{\min}(A\Delta)\geq\sigma_{\min}(A)\sigma_{\min}(\Delta), \sigma is a singular value Let A and \Delta be square matrices. The definition of smallest singular value of a matrix A. (in title, \sigma_{\min}): The matrix norm is the 2-induced norm. The propertie: I don't ... 2answers 45 views ### Prove that \bigg\| \begin{bmatrix} X \\ A\end{bmatrix} \bigg\|_2 \leq \sigma \iff X^* X + A^* A \preceq \sigma^2 I Prove that$$\bigg\| \begin{bmatrix} X \\ A\end{bmatrix} \bigg\|_2 \leq \sigma \iff X^* X + A^* A \preceq \sigma^2 I$$Here, * denotes the conjugate transpose. This norm is the 2-induced ... 0answers 19 views ### Norms (eigenvalues) of sums of orthogonal matrices Let T_1, \ldots, T_n be a set of real-valued symmetric matrices satisfying Tr(T_j T_k) = 0 for all j\neq k. Consider the norm \|T\|_{\infty} = \max_{\|M\|_1 \leq 1} \operatorname{Tr}\left[M^T ... 1answer 38 views ### Finding the global minimum Let f~:~\Bbb R^2\to \Bbb R be defined as:$$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 ...
In the paper "On best approximate solutions of linear matrix equations", there is a very small equivalence I don't know where it comes from. Let $A$ be a matrix (either real or complex), and $\|A\|$ ...