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

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Calculating Cosine Similarity with Matrix Decomposition (matrix multiplication with normalized columns)

To calculate the column cosine similarity of $\mathbf{R} \in \mathbb{R}^{m \times n}$, $\mathbf{R}$ is normalized by Norm2 of their columns, then the cosine similarity is calculated as ...
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Operator norm on $M_n(A)$ where $A$ is a Banach algebra

On $M_n(\mathbb{C})$, if we take the operator norm by acting on $\mathbb{C}^n$ where $||(z_1,\ldots,z_n)||=\max_{1\leq i\leq n}||z_i||$, then for $[a_{ij}]\in M_n(\mathbb{C})$ we have ...
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51 views

A Real Matrix, its Kernel and Image

This is an old exam problem: For an $m \times n$ real matrix $A$, define $\ker A = \{x \in \mathbb{R}^n \mid Ax=0 \}$ and $\operatorname{Im} A = \{Ax \mid x \in \mathbb{R}^n \}$. Show that for all $b ...
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How can we check that for a given norm, we can found an inner product?

Let $$\Bbb C^2=\{w=(z_1,z_2) : z_1,z_2\in\Bbb C\}$$ be the vector space of all ordered pairs of complex numbers. Can we obtain the norm defined on $\Bbb C^2$ by $$||w||=|z_1|+|z_2|$$ from an inner ...
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41 views

Can I solve a problem like a combination of PCA and compressed sensing?

$$ \underset{A,x}{\text{minimize}} \quad \lambda \left\| x \right\|_{1} + \left\| A \right\|_{*} $$ $$ D = A + Mx $$ Where $M \in \mathbb{R}^{n \times m}$, $x \in \mathbb{R}^{m \times z}$, $E=Mx \in ...
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4answers
130 views

Show that $A^{n}\to0$ if and only if $\|A\|^{n}\to0$

Let $\boldsymbol{A}$ be a square matrix. Show that $\lim_{n\to\infty}\boldsymbol{A}^{n}=0$ if and only if $\lim_{n\to\infty}\|\boldsymbol{A}\|^{n}=0$ for the spectral radius or for some operator ...
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38 views

Finding a metric such that $\Phi$ becomes a contraction

Let $$ f:\:[0,1]\rightarrow\mathbb{R};\\ \Phi: \mathcal{F}([0,1], \mathbb{R}) \rightarrow \mathcal{F}([0,1], \mathbb{R}),\\ f\mapsto\Phi(f):=Φf := \left(x\mapsto\left\{ ...
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Equivalence of norms in $C^1[0,1]$

i have the following problem/questions: I have to prove that $\lVert \cdot \rVert_1 \sim \lVert \cdot \rVert_{*} $ in $C^1[0,1]$; Where $\lVert \cdot \rVert_1$ is the usual $C^1[0,1]$ norm and ...
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Spectral radius and matrix norm inequality as its consequence

I am trying to undestand a proof and there is one part that's holding me back. By assumption we have that spectral radius $\rho(A) < 1$. Hence, following inequality should hold $$\|A^k\| < C ...
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17 views

Matrix factorization inequality

How does one show that the following matrix factorization inequality holds in $M_{n} (\mathcal{A})^{+}$, $$(a_{i}^{*}a^{*}aa_{j}) \leq ||a^{*}a|| \cdot (a_{i}^{*}a_{j})$$ Notation. Let $M_{n} ...
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21 views

Derivative with respect to a 2-norm

Given $f = f(u(t), t)$, is it possible to find the following derivative? ($f$,$u \in \mathbb{R}^n$, $t \in \mathbb{R}$) $$ \frac{\partial f}{\partial||u||^2_2} $$ I am aware of the following ...
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How do I find $\|T\|$ when given a matrix $T$?

How do I find the norm $\|T\|$ of T: $\mathbb{R}^3 \rightarrow \mathbb{R}^3$ is defined by $T(x) := Ax$, where $A:= \begin{pmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 3 ...
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1answer
38 views

Find the norm of operator

Let we have the following operator $$T:\ell_p \rightarrow \ell_p$$ $T(ξ_1,ξ_2,ξ_3,ξ_4,ξ_5,ξ_6,ξ_7,ξ_8,ξ_9,ξ_{10},ξ_{11},ξ_{12},ξ_{13},ξ_{14},..)=(ξ_1,ξ_3,ξ_5,ξ_7,ξ_9,ξ_{11},ξ_{13},...)$ How can I ...
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Does unitary transformation preserves the max of norm-2 of lower dimensional vectors?

Let $\{\mathbf{e}_i\in\mathbb{R}^n, i=1,...,N\}$. Apply a unitary transformation of the form $\mathbf{U}_N\otimes\mathbf{I}_n$ to this vector set and reach to vector set ...
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orthogonality-like in vector normed space

Given a normed real vector space $V$, and a vector $x\ne 0$, there exists a vector $y\ne 0$ such that $\|x+y\|=\|x-y\|$? I know that it exists if the norm is induced by a scalar product (in ...
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52 views

Dense subsets in tensor products of Banach spaces

Assume $B_1$ and $B_2$ are Banach spaces of univariate functions. Moreover, assume that the sets $D_1 \subset B_1$ and $D_2 \subset B_2$ are dense with respect to the respective norms ...
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1answer
39 views

A question on Hölder inequality [duplicate]

Let $p, q > 1$ such that $\frac{1}{p} + \frac{1}{q} = 1$. Then $$|\sum\limits_{i = 1}^n x_i y_i| \leq ||x||_p ||x||_q, \;\; \forall x, y \in \mathbb{R}^n.$$ I have to prove it considering $$u = ...
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1answer
14 views

Prove that there is an inner product on $\mathbb{R}^2$, given that the associated norm is a p-norm only if p = 2

Prove that there is an inner product on $\mathbb{R}^2$, such that the associated norm is given by: $ \parallel (x,y) \parallel = (|x|^p + |y|^p)^\frac{1}{p}$ where $ p > 0 $ only if $ p = 2 $ So ...
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1answer
37 views

Functional analysis: $\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}$

Let $(X,\langle\cdot ,\cdot\rangle)$ an inner product space and $A\in\mathcal L(X)$. I have to show that $$\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}.$$ The fact that $\|A\|\geq ...
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How much similar two large matrices are-a practical approach

I'm wondering is there any method to check how much similar two matrices are? For example the following three matrices $A = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right],B = ...
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1answer
20 views

Tightness of inequalities for various matrix norms

For a general inequality involving matrix norms, does the choice of the norm influence the tightness of the inequality? Eg. In $\|AB\| \leq \|A\| \|B\|$, Does the choice of the norm affect the ...
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1answer
19 views

Strict inequality of vector norms

Given a non orthogonal projection $p$ and non zero vector $x$. I am going to prove that $$\|Px\|<c\|x\|$$ for some $c<1$, where $\|\cdot\|$ is the usual Euclidean norm. I can only have the ...
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26 views

matrix operator norm and inner product

Is it true that $\Vert A\Vert:=\sup_{\Vert x\Vert=1}\Vert Ax\Vert=\sup_{\Vert x\Vert=\Vert y\Vert=1}\vert\langle y,Ax\rangle\vert$ for arbitrary matrices $A$? Showing $"\geq"$ seems to be ...
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47 views

Completeness in $L^p$ and supremum norms

As seen by this question, $C([0,1])$ is not complete in the $L^p$ norm ($1 \le p < \infty$) because one can choose a Cauchy sequence $$f_n(x) := \begin{cases} 0 & \text{if $0 \le x \le ...
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1answer
71 views

Why is the Euclidian norm used to measure complex numbers?

Why is the Euclidian norm used to measure complex numbers? The complex numbers are numbers (or more precisely, pairs of numbers), and I can't see why are they essentially connected to the ...
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84 views

Taylor series approximation of function under norm

I am reading this paper. At page number $4$, term $||Au - f||^{2}$ is approximated by taylor series approximation around $u^{k}$. The resulting approximations are $$\|Au - f\|^{2} \approx ...
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Norm of the operator $T:\ell^2 \to \ell^2$ defined as $(Tx)_1=0, (Tx)_n=-x_n+\alpha x_{n+1}$

Consider the operator $T: \ell^2 \to \ell^2$ defined as $$\begin{cases} (Tx)_1 = 0, \\ (Tx)_n = -x_n + \alpha x_{n+1}, \quad n\ge 2 \end{cases} $$ where $\alpha \in \mathbb{C}$. I want to find ...
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1answer
45 views

Is there a nowhere differentiable norm on $\Bbb R^n$?

Is there a nowhere differentiable norm on $\Bbb R^n$? The non differentiable norms that I know (e.g. the one norm $\|\cdot\|_1$ and the infinity norm $\|\cdot\|_{\infty}$) are non differentiable ...
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38 views

Find Solution regarding 2-Norm

I try to understand that, but I have no clue what do to and how to do it. $A$ is a $m \times n$ matrix with $rg(A)=m$. Find the solution for $Ax = b$, which is regarding to the $2$-norm (I guess ...
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Do tensor norms exist?

Does there exist norms for tensors, as an extension for the ordinary matrix norm? For example, if there is a derivative of a matrix [A] with respect to a vector {x}, does the norm of this derivative ...
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32 views

How can I find a norm of a linear transformation $T(x,y) = (ax+cy, bx+dy)$?

Let a linear transformation $T : \mathbb{C}^2 \to \mathbb{C}^2$ s.t $T(x,y) = (ax+cy, bx+dy)$ where $a,b,c,d \in \mathbb{C}$. Now, find the norm of T equipped with ($\mathbb{C}^2$ , $l^1(\{1,2\})$ ...
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24 views

Compute the limit of this expression of norms:

Compute the limit, as n goes to infinity, of the quotient: $$\frac{||A^{n+2}(x)||}{||A^n(x)||} $$, given the matrix $$ \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ ...
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Calculating Norms for the Transpose

How does one show that the transpose mapping $T: \mathcal{M}_{n} \to \mathcal{M}_{n}$, given by $T(a)=a^{t}$, has norm $||T||=1$ but completely bounded norm $||T||_{CB}=n$? Notation. ...
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23 views

Taking limits of norms of a matrix raised to the nth power:

Given a matrix $$ A = \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ \end{bmatrix} $$ and a vector $x = \begin{bmatrix}1&0\end{bmatrix}$, compute ...
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4answers
57 views

does $\|A\| > \|B\|$, $A$ and $B$ matrices, imply that $\|Ax\| >\|Bx\|$ for all $x$ in some vector space?

I'm wondering if I can prove the first inequality in a question that I am working on, does that make the second inequality automatic? Thanks,
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1answer
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If every entry of a matrix A is bigger than every entry of a matrix B, is norm A bigger than norm B?

The norm $\lVert A \rVert$ is different from the norm $\lVert A(x)\rVert$, right? Just making sure that I am interpreting questions regarding matrix norm correctly. I am asked to compare the norm of ...
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1answer
16 views

question on projections in linear algebra

Hello all I was given this question for linear algebra class which I am stuck on and would truly appreciate the help: V is a finite-dimensional inner product space with M and N non trivial subspaces. ...
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1answer
30 views

Find a bound for a matrix

Let $A, B \in \mathbb{R}^{n\times n}$ and $I$ be an identity matrix of order $n$. Suppose $$B_k = B_{k-1} + B_{k-1} (I - A B_{k-1}), \quad (k=1,2,\ldots)$$ If $\Vert I - AB_0 \Vert = c < 1$, then ...
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Surface area of a torus

One can generate a torus as follows: $\vec{g}=((b+a\cos u)\cos v, (b+a\cos u)\sin v, a \sin u)$. To find its area, we can use a surface integral of the form $S=\int\int_{D_{uv}} {\lVert \frac{∂g}{∂u} ...
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Let $V$ be an inner product space. Show that if $||x+y||=||x||+||y||$, then $ax=by$ where $a,b$ are non-negative and not both zero.

Let $V$ be an inner product space. Show that if $||x+y||=||x||+||y||$, then $ax=by$ where $a,b$ are non-negative and not both zero. I know that the converse is true. I considered the square of the ...
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Norm of the inverse of a tridiagonal

Let's take a tridiagonal matrix (in general not Toeplitz, nor symmetric) $$L=\begin{pmatrix}a_1 & -b_1 & & & \\ -c_1 & a_2 & -b_2 \\ & -c_2 & \ddots & \ddots\\ ...
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1answer
24 views

Relationship between the square of the Frobenius norm and the Frobenius norm of the square

I am looking to understand the following relationship: I have a matrix $A$, whose entries are all bounded by $0 \leq a_{i,j} \leq 1$, and follows the constraint $\|A\|_2 = 1$. Is there anything ...
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Cauchy sequence which does not converge example.

Consider the normed space $(X, \Vert \cdot\Vert) $ where $$ X=\{ (a_n)_n \quad|\quad (a_n)_n \text{ real sequence with } \lim_{n\to \infty}a_n=0 \} $$ and $$\Vert (a_n)_n\Vert:= \sum_{n\geq ...
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57 views

Understanding Norms On a Vector Space (Part II)

This question is motivated by a previous question of mine. Let $\|\cdot \|$ be any norm(not necessarily the standard norm) on $\mathbf R^3$ and $S$ be the set of all the points with norm $1$. Let $p$ ...
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1answer
32 views

Absolute value of vector not equal to magnitude of vector

I've come accross the following inequality for a norm (where the norm defines the length of the vector): $$\lvert x \rvert ≤ \lvert \lvert x \rvert \rvert \leq \sqrt{n} \lvert x \rvert$$ where $x$ ...
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54 views

Understanding Norms on Vector Spaces

Let $\|\cdot\|$ be a norm (not necessarily the standard norm) on $\mathbf R^2$ and $S$ be the set of all the vectors $v$ such that $\|v\|=1$. For any point $p\in S$, let $\ell_p$ denote the line ...
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27 views

Why can the function $f(x)=||A\vec{x}-\vec{b}||^2$ be rewritten as $\vec{x}^tA^tA\vec{x}−\vec{x}^tA^t\vec{b}−\vec{b}^tA\vec{x}+||\vec{b}||^2$

Someone answered a question introducing this transformation of the function $f(x)=||A\vec{x}-\vec{b}||^2$ ; but I cannot get the idea why and how. Looks a bit like a binomial expansion, but I can't ...
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2answers
27 views

Proof of basic Inner Product Property

I am searching for a proof of the following property for two vectors x and y in $\mathbb R^n$ $<x,y> = ||x||*||y|| \implies ||x||= \lambda*||y||$ for some $\lambda\in\mathbb R$ with ...
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6answers
75 views

Why does this set a compact?

Consider the following set in $\mathbb{R}^n$: $$S = \{ x \ :\ \|x\| = 1 \}$$ Why is this set a compact? I don't know about closed, but it doesn't look bounded to me.
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1answer
71 views

Name for norm with property $\|x+y\|=\|x\|+\|y\|$.

Is there a name for a norm with property $\|x+y\|=\|x\|+\|y\|$ ? I found this, but it doesn't seem to answer my question.