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

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Expression for Lp Norm [duplicate]

Use Holder inequality $(|\sum_{i=1}^n x_n\cdot y_n| \le ||x||_p\cdot ||y||_q)$ to prove that for each $x\in \Bbb R^n$: $$||x||_p=\sup_{||y||_q\le 1} {|\sum_{i=1}^n x_n\cdot y_n|} $$ tried to find for ...
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Finding a Unit Vector v for a Matrix A such that the 2-norm of AV is equal to the 2-norm of A

I have been working on the following problem: Let A be the following 2x2 matrix: A = [1 1; 0 1] (MATLAB notation) Find the 2-norm of A and a unit vector v such that the 2-norm of Av = the 2-norm of ...
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bounding operator norm by quadratic forms on orthonormal basis

Let $\{u_i\}_{i=1}^n$ be a set of orthonormal basis and $M$ a symmetric matrix. Suppose \begin{equation} \left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n. \end{equation} Can I ...
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How to prove $||A||_2\leq ||A||_F \leq \sqrt{n}||A||_2$ [duplicate]

$A$ is a square matrix with dimension $n$ and $||A||_F$ is Frobenius norm.
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Infinite number of induced norms

While proving that a norm had to come necessarily from a scalar product I have started to wonder about the concept and uniqueness of induced norm. My teacher hasn't clarified to me this doubt, saying ...
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Maximum coordinate of a linear transformation of a vector

Given a vector $x \in R^n$ (variable) and a constant matrix $M \in \{0, 1\}^{m \times n}$ (known). $M$ is a binary matrix, meaning that its entries are either $0$ or $1$. I need to obtain an ...
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71 views

Difference: normed space and normed linear space.

I'm currently reading up on Banach spaces. My book "Introduction to Banach spaces and their Geometry" by Beauzamy mentions "normed spaces" in some places, and "normed linear spaces" in other. I really ...
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Norm of a mapping

$$C[0,1]=\{f:[0,1]\rightarrow R | \text{$f$ is continuous function}\}$$ $$\|f\|=\max \{|f(x)|:x\in [0,1]\}$$ $$A:(C[0,1],\|\|)\rightarrow(C[0,1],\|\|)$$ $$A(f)(x)=(x^4-x^2)f(x)$$ I have to ...
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Proving nonsingularity of a sum of matrices

I'm trying to solve this study question but I'm not sure how to proceed. The question is as follows. If \begin{equation}\frac{||B||_2}{||A||_2}<\frac{1}{\kappa_2(A)}\end{equation} with ...
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Extrema of a vector norm under two inner-product constraints.

If $\langle\vec{A},\vec{V}\rangle=1\; ,\; \langle\vec{B},\vec{V}\rangle=c$, then: \begin{align} max\left \| \vec{V} \right \|_{1}=?\;\;\;min\left \| \vec{V} \right \|_{1}=? \end{align} Consider the ...
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279 views

Norm of a vector-valued function?

When studying commutator estimates, I have encountered the following problem. Consider $f\in C^1(\mathbb{R}^d,\mathbb{R})$ with $\nabla f\in L^p$. So $\nabla f(x)\in\mathbb{R}^d$. My question is ...
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1answer
670 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 ...
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Norms for minimizing arbitrary error distribution

Assume we are doing regression on the function $f(x)$ with error term $e(x)$ with distribution $g(e; \theta)$: $ y = f(x) + e(x), \; e(x) \sim g(e; \theta) $ Let's say we know the analytic form of ...
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1answer
49 views

SVM - Min square norm

All Support Vector Machine litterature mentions that optimal hyperplane is found as: max 1/∥x∥ (st. constraints) which translates directly to: min ∥x∥ or equivalently min $ ∥x∥^2 $. Here ...
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Natural invariant norm on the space of polynomials

What is the " Natural" invariant norm on the space of polynomials in a complex variable $z$? And can anyone give me an idea as to how it is deduced?
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Formulas involving the square of a norm

Why does $$\|x-y+\alpha z\|^2=\|x-y\|^2+2\alpha\langle x-y,z\rangle+\alpha^2 \|z\|^2$$ but $$\|x-z+\theta z-\theta y\|^2=\|x-z\|^2+2\theta\langle x-z,z-y\rangle +\theta^2 \|z-y\|^2?$$ Why is there ...
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Calculation of the sub gradient of the first norm of a matrix

Lets say I have a matrix X and its first norm $||X||_1$. How do I calculate the subgradient of this norm with respect to matrix X itself.
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1answer
216 views

How to expand equation inside the L2-norm?

I want expand an L2-norm with some matrix operation inside. Assume I have a regression $Y=X\beta+\epsilon$. I want to solve (meaning expand), $$\displaystyle\|Y-X\beta \|_{2}^2$$ Should I do: 1) ...
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54 views

Is it general to say “norm” to mean 2-norm when it is on an inner product space?

Let $V$ be an inner product space over $\mathbb{F}$. If one defines $\lVert \bullet \rVert$ as $\sqrt{\langle \bullet, \bullet \rangle}$, then $\lVert \bullet \rVert$ is a norm on $V$. However, if ...
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1answer
75 views

Linear approximation of matrix norm

Given a square matrix $X=[x_1...x_N]$, and can be vectorized by $y=vec(X)=[x_1^T ... x_N^T]^T$ Is there any linear function can approximate $|| X ||$ (any matrix norm is okay) by using $y$?
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$V$ is finite dimensional iff $V'$ with the weak topology is normable

Why is the following statement valid? Note, $V$ is locally convex Hausdorff topological vector space over $\mathbb{C}$ and $V'$ is the space of all continuous linear maps from $V \to \mathbb{C}$. ...
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Matrix norm applications?

There'are many different ways to calculate Matrix norm. But once calculated, what is the practical use/application of it (e.g. in computer programming)? Or does it let define something that can be ...
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83 views

All matrix/vector norms induce the same topology?

From Wikipedia all norms on $K^{m \times n}$ are equivalent; they induce the same topology on $K^{m \times n}$. This is true because the vector space $K^{m \times n}$ has the finite dimension $m ...
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53 views

Boundary value problem domain, norm?

I'm starting some research on a boundary value problem for a memoire for the 2nd semester of a masters, and my mentoring professor asked me to find the answer to the following question: For $u \in ...
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1answer
551 views

compute the similarity between two vectors

Euclidean distance is a measure that may be used to compute the similarity between two vectors. Given a query $q$ and documents $d_1, \ldots, d_n$, we may rank the documents $\mathcal{D} = ...
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154 views

Minimum of Frobenius norm

I'm not sure if this solvable. I would like to find an orthogonal matrix that minimizes the following Frobenius norm: $$\min||BQ||^2 \\ \text{s.t. } Q^TQ=diag(\alpha_1,...,\alpha_n)$$ where ...
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21 views

Matrix $L_1$ norm derivative [duplicate]

Say that $A\in \mathcal{M}_{n,n}$ what is the result of the following derivative: $\frac{\partial \|A- diag(A)\|_1}{\partial A}$, where $diag(A)$ is the matrix that contains the diagonal entries of ...
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77 views

Lower bound on the norm of product of non square matrices

The following inequality is known: $\parallel AB\parallel\geq\parallel A\parallel \sigma_{n}(B)$. However, it is only valid where both $A$ and $B$ are square. Is there an analogue for rectangular ...
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Which of the following functions are norms? [duplicate]

For $x=(x_1,x_2)$, which of the following functions on $\mathbb{R}^2$ are norms? a.) $A_1(x) = 7\mid x_1\mid + 3\mid x_2\mid$, b.) $A_2(x) = \text{max}\lbrace\mid x_1\mid^2,\mid x_2\mid^2\rbrace$, ...
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60 views

Approximating asymmetric matrix's Ky Fan norm

Given an N-by-N asymmetric matrix $M$ Is there any theory about approximating $M$'s Ky Fan k-norm $|| M ||_k$ using $\frac {(M+M^T)}{2}$'s Ky Fan k-norm $|| \frac {(M+M^T)}{2} ||_k$? UPDATE: ...
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Expectation inequality with weight matrix

Given that $X$ is random matrix ($X'$ is transposed matrix) and $\Omega$ is invertible square matrix is inequality $$EX'X\cdot(EX'\Omega X)^{-1}\cdot EX'X\leq EX'\Omega^{-1}X$$ correct? All the ...
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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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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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176 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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102 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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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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63 views

A basic question related with the order of convergence of iterative method

I am working on an iteration problem for computing inverse of a non singular matrix $A$ I have got following relationship between error matrix defined by $E_k = X_k-A^{-1}$. $\|E_{k+1} \|\leq ...
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168 views

Supremum of constrained $L_1$ norm

For a fixed $\mathbf{h}$ in a subset of $\mathbb{C}^m$ such that $\mathbf{h}(k)\neq 0$ for any $k=0,...,m-1$, how can I find $\sup_{\mathbf{x}} \{ \| \mathbf{x} \|_1 \,\,\, \mathrm{ s.t. } \,\,\, ...
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182 views

The geometry of functions $\mathbb{R}^2\rightarrow \mathbb{R}$ that satisfy the norm axioms

What are constraints on the "looks" of a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$, if $f$ satisfies $$f(x_1+x_2)\leq f(x_1) + f(x_2), \ \quad x_1,x_2\in \mathbb{R}^2 \quad \quad (1)$$i.e. the ...
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1answer
408 views

Inequality with matrix norms

Prove that if $A$ is invertible and $||A-B||<||A^{-1}||^{-1}$ then $$\lVert (I-A)^{-1}\rVert \leq \frac{\lVert I\rVert-(\lVert I\rVert-1)\lVert A\rVert}{1-\lVert A\rVert}.$$ This is the second ...
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1answer
24 views

Metric induced from norm

I was trying to understand the following: Every norm on $R^n$ is continuous (as a map from $R^n$ to $R$). Proof. We use the maximum metric on $R^n$: $ d(x, y) = \max{|x_j − y_j| : j ∈ \{1, . . . ...
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Multivariable calculus - find total derivative

I want to find the total derivative of the function $f: \mathbb R^n \to \mathbb R^n$, $f(x)=\frac{x}{|x|}$ If I was to copy what the teacher taught, I should find the limit of $\lim_{t \to 0} ...
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In a normed space, is it always true that $\|a_1e_1+\dots+a_ne_n\|\geq |a_i|\|e_i\|$?

In a normed space, is it true in general that $\|a_1e_1+\dots+a_ne_n\|\geq |a_i|\|e_i\|$ for all $1\leq i\leq n$? $e_i$ are basis elements of the vector. This is definitely true for the Euclidean ...
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1answer
558 views

Sobolev space - norm $H^1$ and $H^1_0$

When we defined on $H^1_0$ the norm $$||v||_{H^1_0}=||v||_{L^2}+||\nabla v||_{L^2}$$ can we tell that $$||u||_{H^1_0} = ||u||_{H^1}?$$ Thank's
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Is delta distribution continuous and differentiable with dual space norm?

I know that delta distribution $\delta : \mathcal S (\mathbf R) \to \mathbf C$ is continuous with usual seminorm and here. I am interested in its continuity with dual-space $H^{-1}(\Omega)$ of ...
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derivation of vector norm

what would be the differentiation of this equation :- $F(A) = \sum_{i} \left \| Y_{i} - AB_{i} \right \|^{2} + \lambda \left \| A - C \right \|^{2}$ wrt to A . Y is a column vector and B is column ...
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Norm inequality? [closed]

What are best possible constants $K_1,K_2,K_3$ would the inequality \begin{align} K_1||f||_2||g||_2 \le K_2||f||_p||g||_q \le K_3||f||_1||g||_{\infty} \end{align} holds? for all $p>1$ with ...
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1answer
75 views

Contraction map in extending domain from Dense subset to $L^{2}$

This thread is about extending a dense domain $D \subset L^{2}$ into $L^{2}$. I do not understand what Deyton means in his comment about getting contraction map when doing this. I cannot see any ...
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Conditional number: exercise

Let's say we've got a $202 \times 202$ matrix $A$ for which $||A||_2=100$ and $||A||_F=101$ (the Frobenius norm). How can we find the sharpest bound (lower) on the 2-norm condition number of $A$? ...