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

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Norm of a Matrix-vector product

Suppose I have vector $\vec x \in \mathbb R^n$ and matrix $\mathbf M$ of dimension $m\times n$. Is there an alternative expression for $\lVert \mathbf M \cdot \vec x \lVert$ that includes $\lVert \vec ...
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What is the Hessian of Frobenius norm

As we know that every norm is convex, and if a function is convex w.r.t. the input variable, then corresponding Hessian should be positive semidefinite. When I try to find the Hessian of Frobenius ...
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Derivatives of Norms and Absolute Values (distributions)

For example we have for $x \in \mathbb{R}$, $$\frac{\partial}{\partial x}\left| x\right| = 2\Theta(x) -1 $$ and thus $$\frac{\partial^2}{\partial x^2}\left| x\right| = 2\delta(x) $$ We also have, ...
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What minimizes the Chebyshev Distance?

For an arbitrary number of dimensions, I know that the mean minimizes the distance using the L2 norm and that the geometric median minimizes the distance function using the L1 norm (though I have yet ...
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What is the relation between Chebyshev Norm and L1 norm in $D$ dimensions

How do you translate one to the other (Chebyshev norm to the L1 norm) for $D$ dimensions?
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Distance metrics with kmeans

Context: I'm trying to derive some formulas for computing the "mean" in the K-means algorithm. So given an assignment of $m$ data points to $k$ clusters, find a formula to recompute the mean of the ...
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Inequality of Frobenius norm for skew matrices

Let $A$ be a complex skew-symmetric $n \times n$ matrix, that is, $A^T = -A$. Denote by $\|\cdot\|_F$ the Frobenius norm, that is, $\|B\|_F^2 = \text{trace}(B^*B)$. I would like to prove that $$ ...
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What is the usual norm of $\mathbb{H_0^1}$?

What is the usual norm of product of sobolev spaces $\mathbb{H_0^1}=H_0^1 \times H_0^1=W^{1,2}\times W^{1,2}$? In my work i need to prove that the norm endowed by the inner product ...
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Equivalence of norm in a ring.

For the ring $$A=\mathbb{Z}[i\sqrt{3}]=\{a+i\sqrt{3}b:a,b\in \mathbb{Z}\}$$ I had to show that the only invertible elements are $1$ and $-1$, using the norm $$N:\mathbb{Z}[i\sqrt{3}]\quad ...
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Prove scalar product in a normed vector space is an open mapping.

I have the feeling this is a really obvious question, but I'm having trouble with it, here it goes: Let $(X, ||\,||)$ be a normed vector space over $K$, prove that $\odot:K\setminus\{0\}\times X ...
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Prove relative error with condition number of matrix inequality

I was working on some questions and solutions, and encountered the following question. I am able to prove the inequality using the given information and some algebraic manipulation but the "within ...
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Proving that derivative of a bounded linear map (at a point) is the map itself

I have been struggling to prove a claim. How can I show that If $f:X \rightarrow Y$ is a bounded linear map, then $Df(x)=f$ for all $x \in X$? Attempt: $f:X \rightarrow Y$ is a bounded linear map ...
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Norm of linear functional

For every $ x \in C([a,b])$, we define the functional $F(x) = \sum_{i=1}^{n} {\lambda_{i} x(t_{i})}$ where $\lambda_{i} \in R, i=1,...,n$. I was wondering if someone can help me to find a sequence ...
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Check my answer - Finding the jacobi matrix of a function

We are given $f: \mathbb R^n \to \mathbb R^n$ such that: $0 \neq x \in \mathbb R^n$, $f(x)=\frac{x}{|x|}$, where $|x| = \sqrt {x_1^2 +x_2^2+...+x_n^2}$ Find the jacobi matrix (the differential ...
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Prove this result about norm 2 of a matrix

$\left | \left | A \right | \right |_{2} :=\left ( \sum_{i,j=1}^{n} a_{ij}^{2} \right )^\frac{1}{2}$ for $A\in \mathbb{R}^{n\times n}$ Show this: If A is a simetric matrix then $$\left | \left | A ...
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Condition of a matrix proof

I have to show the following inequality: For given two invertible matrices $A,B \in \mathbb{R}^{n\times n}$ show that $k(AB)\leq k(A)k(B)$, where $k(A)=\left \| A \right \|\left \| A^{-1} \right \|$ ...
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Special Matrix 2-norm and F-norm Inequalities

This is a homework problem for my Numerical Linear Algebra course. It states the following: If A is an mxm nonsingular matrix, prove the following: (1)$\|A+(A^{*})^{-1}\| _{2} \ge 2$ ...
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Derivative of nuclear norm

I'm trying to take the derivative of nuclear norm with respect to its argument. nuclear norm is defined in the following way: $$\|x\|_*=\mathrm{tr}(\sqrt{x^Tx})$$ I'm trying to calculate: ...
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Submultiplicativity stronger than triangle inequality?

I would like to ask a question about matrix norm. Is the submiltiplicativity property always stronger than the triangle inequality? So, if i prove for a matrix norm that it's submultiplicative, i ...
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proof for matrix norms

How do I prove these two inequalities on matrix norms: $\Vert A \Vert_1 \leq n\Vert A \Vert_\infty,$ $\Vert A \Vert_1 \leq \sqrt{n}\cdot\Vert A\Vert_F$ , where A is $m$-by-$n$ real matrix.
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Why is the determinant function continuous with regards to the Hilbert-Schmidt norm on matrices?

Why is the determinant function continuous with regards to the Hilbert-Schmidt norm on matrices? I know that the determinant is polynomial of the elements of the matrix, and since $\|A\|_{HS}^2 = ...
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Norm of an infinitesimal Vector

Supose I have the following displacement vector: $$ \Delta\vec y(x) = (\Delta y_1(x), \Delta y_2(x), \cdots, \Delta y_n(x)) $$ And say we define the unit vector $\hat n$ as: $$ \hat n = ...
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Matrix norm proof

Given is $\left | \left | A \right | \right |_{2} =\left ( \sum_{i,j=1}^{n} a_{ij}^{2} \right )^\frac{1}{2}$ for $A\in \mathbb{R}^{n\times n}$. Show that this defines a matrix norm. I remember i've ...
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Does $f_n$ converge to $f$ uniformly?

Consider $f_n: [0,1]\longrightarrow\mathbb{R}$ is given by $$f_n(x) = \begin{cases} \sqrt{n}, & \quad 0<x<\frac{1}{n} \\ 1, & \quad \text{otherwise.} \end{cases}$$ 1.) ...
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Derivative of a Vector with respect to its norm (special relativity)

I came across an equation (related to special relativity) that requires me to to take a derivative of a vector with respect to to it's own norm. In a bit more detail, what I mean is, let: $$\vec ...
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Natural matrix norm of an inverse matrix

Let $\left\|\cdot\right\| : \text{GL}(n,\mathbb{R})\to\mathbb{R}_{\ge 0}$ denote the natural matrix norm, i.e. $$\left\|A\right\|:=\max_{x\ne ...
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Inequality for norm of linear combination of linearly independent vectors

I'm trying to find a proof for the following: Let {$u_{1},...,u_{n}$} be a linearly independent set of a normed space $X$. Then, there is a constant $c>0$ such that for every set of scalars ...
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How does one prove $\left\lVert\sum\limits_{n=1}^{\infty}f_n\right\rVert\le \sum\limits_{n=1}^{\infty}\lVert f_n\rVert$?

Can the triangle inequality for norms works for infinite some as well? Hope that someone can give some hints to prove it or give a counterexample. Thanks!
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Let $f \in L^1[0,1]$, can we approximate $f$ by a polynomial, in sup norm?

Let $f \in L^1[0,1]$, can we approximate $f$ by a polynomial, in sup norm ? I know that the algebra of polynomials is dense in algebra of continuous functions, wrt to sup norm, And I know that if $f ...
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Comparable norms on the space of polynomials?

Are the norms: $$\|P\|_1=\int^1_0\|P(t) \| dt\mbox{ and }\|P\|_2=\sup_{0\le t\le1} |P(t)|.t$$ comparable on the vector space $X=\mathcal{P}[0,1]$, i.e. all polynomials on $[0,1]$? Here, i try to ...
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Prove that the operator norm is a norm

Exercise: Prove that the operator norm of the set $S$ of all linear operators $L:R^n\to R^m$ defines a norm on $S$ Definition of norm: A positive function $\| .\|$ on a real vector space $V$ is a ...
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The norm of a vector with gaussian noise

Say I have a vector of length n, $v \in R^n$ where $0<=v(i)<=1$ for each i, Now, I add noise: let $n$~$Normal(0,\sigma)$ And to each i I add noise v(i)+n , such that the noises are independent ...
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Why this relation in Hilbert space (with inner product $< >$) holds?

$c_k$, $f_k$ are sequences in Hilbert space, $g$ is a function. Why this relation below holds? How you derive it? $\sum_k|c_k\left \langle f_k,g \right \rangle|\leq(\sum_k|c_k|^2)^{1/2}(\sum_k|\left ...
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Why $||f||=\mathrm{sup}_{||g||=1}|\langle{f,g}\rangle|$

My question is why $||f||=\mathrm{sup}_{||g||=1}|\langle{f,g}\rangle|$? for any two functions $f$ and $g$ with $||g||=1$, and $||\;||$ denotes the 2-norm. I have tried to use the triangle inequality ...
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Conditions on $\alpha_n$ for $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ to be a norm on $l_p$

When $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ is a norm in $\mathcal{l}_p=\lbrace (x_k)^\infty_1 : \sum\vert x_k\vert ^p \lt\infty\rbrace $ and $\alpha\in\omega$. and $\omega$:space of ...
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If A is positive-definite what can we say about $x^t A y$ or $y^t A x$?

If $A$ is an $n \times n$ positive definite matrix in $\displaystyle f(x) = \frac{\sqrt{x^t A x}}{2}$, can I claim that $f(x+y) \leq f(x) + f(y)$?
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Relation between the weighted matrix norm and the weights

For a nonsingular matrix $W \in \mathbb{C}^{m\times{}m}$, the weighted vector norm is defined as $||\overrightarrow{x}||_W = ||W\overrightarrow{x}||$. Let $||A||$ denote the induced matrix norm by the ...
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Can anyone explain this isometry to me? $T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty})$, $ T(x)(y) = \sum_{i=1}^n x_i y_i$

Can anyone explain this isometry to me? $$T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty}),\qquad T(x)(y) = \sum_{i=1}^n x_i y_i$$ I don't get what the domain and image of $T$ are. ...
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proofread $\left \| f -g\right \|_{p}^{p}=p(p-1)\int_{0}^{\infty}\int_{0}^{t}[m(f>t/g>s)+ m(g>t/f>s)]|t-s|^{p-2}dsdt$

Is this argument correct? $\left \| f -g\right \|_{p}^{p}=$ $\int_{0}^{\infty}m(|f-g|>t^{\frac{1}{p}}) dt\stackrel{c.o.v.}{=}$ $p\int_{0}^{\infty}m(|f-g|>t) t^{p-1}dt\stackrel{t=|u-s|}{=}$ ...
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Can we find equivalent norms for ordinary numbers $m$ and $M$ s.t. $0<m<M$?

Let $\|\cdot\|$ be a norm and $0<m<M$. Can we say that we can find another norm $\|\cdot\|_1$ s.t. $\|\cdot\|$ and $\|\cdot\|_1$ are equivalent with respect to numbers $m$ and $M$?
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Is the norm $\|A\|_\mathrm{op}$ for a matrix $A$ a Euclidean norm?

$\displaystyle \|A\|_{\operatorname{op}} = \sup \frac{\|Av\|}{\|v\|}$ for $v$ in $V$, where $\|v\|$ is a Euclidean norm of a vector $v$ in a vector space $V$. I saw a counter-example that was based ...
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For a matrix $A$, is $\|A\| \leq {\lambda}^{1/2}$ true?

In class I saw a proof that went something along these lines: Define $\|A\| = \sup \dfrac{\|Av\|}{\|v\|}$ for v in V, where the norm used is the standard (Does this even exist?) Euclidean norm in V. ...
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Computing H1 norm numerically

I'm solving a PDE numerically using FDM and Spectral Methods. I understand how to compute the $L_{2}$, but I dont understand how to compute the $H_{1}$ norm. What does the $u'$ mean in the below ...
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let $A$ be an n by n matrix, show that $||A||_{OP} \leq ||A||_{HS} \leq \sqrt{n} ||A||_{OP}$

We are given $A \in M_{n}(\mathbb R)$ and the following norms: $||.||_{e}$ is the standard euclidean norm of $\mathbb R^n$. $||A||_{OP}$ is the operator norm of $A$, meaning $||A||_{OP} = ...
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37 views

Proof check for completeness

I'd like to see if the proof I have is adequate. Statement. Let $X$ and $Y$ be Banach space, the product $X\times Y$ is a vector space under coordinate operations with norm $$ \|(x,y)\| = \|x\|_X ...
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An equation related to covariance matrix, square root of the matrix, and Euclidean norm.

How can I prove this equation: $${ ({ x }^{ T }\Sigma x) }^{ 1/2 }={ \left\| { \Sigma }^{ 1/2 }x \right\| }_{ 2 }$$ In which $\Sigma $ is a covariance matrix. I tried some numerical examples in ...
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When sup norm,i.e. $\| x|| = \sup|x(t)|$ for $\forall t\in T $in C[0,1] for $T \subsetneq [0,1]$

When sup norm, i.e. $\| x|| =\sup|x(t)|$ for $\forall t\in T $in C[0,1] $T$ is such that $T \subsetneq [0,1]$. What condition should be applied to $T$ to make $\| x||$ a norm. I cannot show for ...
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37 views

Continuous functions. Second norm

Let $f:[a,b]\rightarrow \mathbb{R}^{d}$ be continuous. I need to prove that $\left \| \int_{a}^{b}f(x)dx \right \|_{2}\leq \int_{a}^{b}\left \| f(x)) \right \|_{2}dx$.
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22 views

Analytical solution to fitting two functions

I have two oscillatory functions $f(x)$ and $(k x)^2 g(x)$ where $f$ and $g$ are known and it is also known that the two functions are approximately similar. How can I analytically find the best ...
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bounded subset of normed space

Suppose $X$ is a normed linear space and $S\subset X$. Show that if $$\sup_{x\in S}\{\mid f(x)\mid\}<\infty$$ for every $f\in X^{\ast}$, then $S$ is a bounded in norm.