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

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Definition of renorming of a space

The following is Proposition $4.5$ from Kalton's paper. Let $X$ and $Y$ be Banach spaces such that there exists a Lipschitz embedding $L:X \rightarrow Y$ such that $L(0)=0$ and ...
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31 views

L2 norm of 2 Normally distributed variables

Given: $Z=\sqrt{X^2+Y^2}, X\sim N(\mu_x,\sigma_x^2), Y\sim N(\mu_y,\sigma_y^2)$ What is the expected value of $Z$? I'm specifically looking for the case where the $\mu_i$ are non-zero and $\sigma_i$ ...
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52 views

Dot product and a norm

Let $\langle\cdot,\cdot \rangle$ be a dot product on $\mathbb{R}^{2}$. We define a norm $\|x\|=\sqrt{\langle x,x \rangle}$. We know that: $$ \sup_{x \in \mathbb{R}^2}{\frac{\| x\|_2}{\|x\|}}=3 ...
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13 views

Two-norm function norm of interpolation

I am attempting to calculate the two-norm of this function $f_k(x)$. $$ \|f_k\|_2 = \left( \int_0^1 f_k(x)^{2} dx \right)^{1/2} $$ The answer is listed as $$ \sqrt{2/3} $$ However, I'm not ...
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$\|A\begin{bmatrix} m\\n \end{bmatrix}\| = \|B\begin{bmatrix} m\\n \end{bmatrix}\| $, what can be said about A and B?

Given two 2x2 matrices A and B such that for some real m and n: $$\|A\begin{bmatrix} m\\n \end{bmatrix}\| = \|B\begin{bmatrix} m\\n \end{bmatrix}\| $$ The norm is euclidean, and $A\ne B$ and $m\ne 0$ ...
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60 views

Derive the dual function $g(\lambda, \nu)$ for the least-norm problem

I am trying to find the dual function $g(\lambda, \nu)$ to this problem $$\min\limits_{Ax = b} \|x\|$$ Step 1. Form the Lagrangian $$L(x, \lambda, \nu) = \|x\| + \nu^T(Ax-b) = \|x\| + \nu^TAx - ...
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21 views

Am I showing the positive-definiteness of a symmetric matrix A correctly?

A is a 4x4 square, symmetric matrix. First I computed its Cholesky Decomposition $LL^t$ since the first part of the question asked for it. Then it asked to show A was positive-definite. I am a ...
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22 views

Orthogonal matrices and different norms

I'm trying to understand the relation between orthogonal matrices and different norms through a few practice problems, but as they don't have a solution guide I've gotten particularly stuck on one. ...
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67 views

$|Ax|≤\|A\||x|\space\forall x\in\mathbb{R}^n$ (Rudin's Principles)

In Rudin's Principles of Mathematical analysis p. 208 $\|A\|$ is defined as the $\sup$ of all numbers $|Ax|$, where $x$ ranges over all vectors in $\mathbb{R}^n$ with $|x|≤1$. Then he claims ...
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Find norm of operator $L(x,y)=(x+3y,y-x)$

I'm trying to tackle the following question, but with no success... Let $L: \ \mathbb{R^2}\to\mathbb{R^2}$ be an operator such that $L(x,y)=(x+3y,y-x)$. Find $\|L\|$. So, I know that I need ...
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Why is $\max_{1≤i≤m,1≤j≤n}|a_{i,j}|≤||A||_{op}$

Why is $$\max_{1≤i≤m,1≤j≤n}|a_{i,j}|≤||A||_{op}$$ where $$||A||_{op}=\sup\{||Ax||\space |\space x\in\mathbb{R^n}, ||x||=1\}\space\space\text{(operator norm)}$$ ?
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20 views

arg min-invariance for norm of vectorfield under linear transformation

Given a vectorfield $\vec{F}(\vec{c}) \in \mathbb{R}^n$ which is a function of some parameters $\vec{c}$, what constraints must you have on a matrix such that when you act on the vectorfield the ...
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1answer
54 views

How can I prove this proposition of linear algebra?

Good afternoon! I have to show this proposition: 1) Let $A \in \mathbb R^{n\times n}$ a non singular matrix and $PA=LU$, $P$ permutation matrix, $L$ a lower triangular matrix with $1$ on its ...
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37 views

“Decay” of $L^2$-norm of the solution of heat equation with mixed boundary conditions

I'm considering the heat equation $$u_t = u_{xx}$$ on the interval $[0, \pi]$ with mixed boundary conditions $$u(0,t)=0 \quad \text{and} \quad u_x (\pi,t)=0, $$and smooth initial data $u(x,t)=u_0 (x)$ ...
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37 views

The norm of $A(A^HA)^{-1}A^H$ [closed]

I see a relation in my research and I want to know how can I prove this: $$\|A(A^HA)^{-1}A^H\|_2=1$$ $H$ is transpose.
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22 views

Min (or Max) $L_1$ norm of Hermitian matrices with given eigenvalues

Suppose the real numbers $(\lambda_1, \ldots, \lambda_n)$ are given. All Hermitian matrices that have eigenvalues $\lambda_i$, $i=1,\ldots,n$, can be represented in the following way: $H = ...
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24 views

Householder QR decomposition

Consider the $p × q$ matrix $A$, with $p > q$ and $rank(A) = q$. Let the Householder-QR decomposition of the matrix A be denoted as $$A = [Q Q_N][R 0_{(p-q)*q}]^T$$ Prove that $$||x||^2_2 = ...
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37 views

The inclusions between unit balls in $\ell^p$ spaces

I need to show that $d_∞(x, y) ≤ d_2(x, y) ≤ d_1(x, y) ≤ nd_∞(x, y)$ where $d_1=|x_1-0|+|y_1-0|$ and I'm setting $|x_1-0|+|y_1-0|<0$. Illustrating the $B(0,1)$ balls (centered at 0 with radius 1) ...
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25 views

What does a norm which is symmetric “around” an index subset look like?

I am looking at norms $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$ that have the following symmetry property $$\forall \beta \in \mathbb{R}^{n} \text{: }, \lVert\beta \rVert=\lVert ...
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1answer
21 views

Householder QR problem

Can somebody give me a hint or help me to solve this problem. Let V be a p×q matrix with orthonormal columns (p > q), and $M = I−2VV^T$ , with I being the p × p identity matrix. The matrix M can ...
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14 views

Norm of a full rank matrix and its transpose

If A is p × q with rank q, prove that $||\ A(A^TA)^{−1}A^T ||\ _2 = 1$. What does $A(A^T A)^{−1}A^T$ represent?
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Norm of orthogonal matrices

Can someone help me with this problem. I have no idea how to solve it!! If A is a p×q matrix, U is a p×p orthogonal matrix, and Z is a q×q orthogonal matrix, prove that $||A||_2=||UAZ||_2$
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30 views

L1-norm minimisation

I am working on this tutorial question. The question asks me to write a Matlab code to implement the method. I was stuck in how to formulate a code for the proximal operator as well as the ...
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1answer
19 views

Norm of a matrix evaluation

In the following A denotes a p×p matrix, and x a p×1 vector. (a) Is $f_1(x) = ||Ax||^2$ a norm on the space of p×1 vectors? What are the conditions (if any) that the matrix A needs to satisfy for ...
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If A is invertible and $||B-A|| < ||A^{-1}||^{-1}$ prove $B$ is invertible.

Just having really hard time trying to proof : If $A$ is invertible and $||B-A|| < ||A^{-1}||^{-1}$ prove $B$ is invertible. It is related to Neumann Series but i don't understand how to proof ...
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247 views

Why is the norm convex?

Why is the norm a convex function?                          
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Condition number of input y and output x

Take y=area of a rectangle with corner vertices [0, 0] and $[x_1,x_2]$, where input is $[x_1,x_2]$, output is y and input norm is $||_2$, output is absolute value. How to calculate the condition ...
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How to design a strict constraint term (never violate this constraint) for matrix decomposition

It is a low-rank sparse decomposition problem: $ D= ML + S + \epsilon$, that we know the matrix D can be decompose into 2 part that one part $L$ is low rank, the other part $S$ is sparse, $\epsilon$ ...
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Confusion about norms

I was doodling at the kitchen table this morning and I seem to have thoroughly confused myself about convex functions and norms. A norm $||x||$ is a convex function, via the triangle inequality. ...
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1answer
28 views

dual pairs in $\mathbb R^n$ with non-standard norms $l_1$ and $l_{\infty}$

Suppose $X=Y=\mathbb R^n$. Usually, to apply a separating hyperplane theorem in $X$ (the primal space), we associate both $X$ and $Y$ with the standard Euclidean norm. My question is that could I ...
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Bounding the spectral norm of the inverse of a matrix sum (useful bounds, tightest not necessarily needed)

Let $A$ and $B$ be $n\times n$ matrices. Suppose $A$ is invertible and that $\|A\|_2,\|B\|_2,$ and $\|A^{-1}\|_2$ are known. Are there any useful bounds (upper and/or lower) $||(A+B)^{-1}||_2$ in ...
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33 views

Can we prove that $\lim \|X^T A X\| / \|X\| = 0$ as $\|X\|\to 0$ for all norms?

Suppose $X$ and $A$ are two matrices of compatible dimensions. Is it possible to prove that $$ \lim_{\|X\|\to 0} \frac{\|X^T A X\|}{\|X\|} = 0, $$ where: A is squared, X may not be squared. The ...
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Law of cosines is equivalent to a dot product identity

In this lecture, starting at around 18:00, it's shown the dot product identity $$a^T b=||a||_2 ||b||_2 \cos \theta$$ where $\theta$ is the angle formed b/w $a,b\in \mathbb{R}^3$, is the same as the ...
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39 views

How to check whether the following functions are norms on the corresponding spaces?

$\|f\|$:=sup$_{x\in[0,1]}$$\quad \frac{\vert f(x)-f(0)\vert}{x}$ on the space C[0,1] and the same function but on the subspace V of $C^1$[0,1] of functions which equal zero at x=$\frac{1}{2}$. How ...
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40 views

Prove by Induction that Norms in a Finite Dimensional Space are Equivalent?

I would have thought that this is a good candidate for an inductive proof, but I have searched for one and failed. Is there such a proof, and if not why not ? Here's how far I got. It's easy to ...
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39 views

$p$-norms inequalities [closed]

I got this math homework and I can't do it. I have to prove the inequalities for p-norms in $\mathbb R^n$: $\|x\|_1\ge\|x\|_2\ge\|x\|_\infty$ $\|x\|_1\le\sqrt{n}\|x\|_2$ (I have already proven this ...
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55 views

Compactness of the unit sphere in finite-dimensional normed vector space

In order to prove that norms defined on any finite-dimensional real (or complex) vector space $E$ are equivalents, I need to proof the compactness of the unit sphere $S_{\infty}=\{x\in ...
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Completeness of the space $C_{2\pi} \left ( \mathbb{R} \right )$ [duplicate]

Denote $C_{2\pi} \left ( \mathbb{R} \right )$ is the set of complex-value, continuous on $\mathbb{R}$, periodic functions with period $2\pi$. Fix $p \in \left [1, \infty \right )$. For each $x \in ...
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soft thresholding derivation with two terms

I am trying to minimize the following function $$argmin_x||x-y||^2_2 + ||x-z||_2^2 + \lambda || \frac{x- w}{c}||_1$$ I have been doing it for while but i am not sure how to do it. I have already been ...
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Embedding of Sobolev space W_2^1[a,b] in C[a,b]

Let's define $$||f||_{1,2}=[\int_a^b(f(x)^2+f'(x)^2)dx]^{\frac{1}{2}}$$ and the Sobolew space $W_2^1[a,b]$ to be the completion of $C^1[a,b]$ with respect to $||f||_{1,2}$ norm. How can we show that: ...
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Norms are not equivalent in $c_0$

Consider two norms in the space $c_0$: $$\lVert x \rVert = \sup \lvert x_i \rvert$$ and $$\lVert x \rVert _0=\sum 2^{-i} \lvert x_i \rvert$$ Prove that above two norms are not equivalent. I know ...
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What does $\mathbb{R}^\mathbb{N}$ actually mean?

We all know what $\mathbb{R}^n$ means. But I came across this statement about $\mathbb{R}^\mathbb{N}$ in a note that says $\mathbb{R}^\mathbb{N}$ is a vector space under pointwise operations has no ...
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33 views

Prove that $\| A \| = \lvert y \rvert$

For all $A \in L(\Bbb R^n, \Bbb R)$ there is a unique $y \in \Bbb R^n$ such that $A\textbf{x} = \textbf{x} \cdot \textbf{y}$. Prove that $\| A \| = \lvert \textbf{y} \rvert$. Hint: Cauchy-Schwarz ...
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41 views

Inequality between 2 norm and 1 norm of a matrix

When reading Golub's "Matrix Computations", I came across a series of norm inequalities. While I could prove a lot of them, this one has me stuck: $$ \frac{1}{\sqrt{m}} ||A||_1 \le ||A||_2 \le ...
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23 views

Distance between vectors multiplied by scalars

Suppose that $x,y$ are some vectors in a Euclidean space and $a,b$ are some scalars. Is there any inequality to factor out $||x-y||$ from $$||a x - b y || $$ like this: $$ ||a x - b y || \leq ...
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Show that $||x||$ is a norm on $\mathbb{R^n}$

Let $A\subset \mathbb{R^n}$ be any limited, open, convex, and the centre symmetry set having centre at 0. Show that $||x|| = \inf \{k>0 : x/k \in A \}$ is a norm at $\mathbb{R^n}$ and open ball ...
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1answer
18 views

$\|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0$ Why is $A$ injective?

Why does $A:X\to Y$ where $X$ is a banach space and $Y$ is a normed space, where $A$ is a surjective bounded linear operator, where: $$\|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0$$ Mean that $A$ is also an ...
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2answers
40 views

Finding the Unit Quaternion

How can i take a Quaternion and find the Unit Quaternion. How can I find the Unit Quaternion (Norm of a Quaternion). The norm of a Quaternion should be equal to $1$ E.g. $a=(2-i+2j-3k)$ Here is what ...
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1answer
23 views

Should you drop the inner absolute value sign for $L2$ norm?

Lp norm is defined as: $ \left\| \mathbf{x} \right\| _p := \bigg( \sum_{i=1}^n \left| x_i \right| ^p \bigg) ^{1/p}$ But often time I see people writing: $\left\| \mathbf{x} \right\| _2 := \bigg( ...
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1answer
46 views

$A$ is normal, and $B$ is Hermitian, why does $\left\| \rm{AB} \right\|{\rm{ = }}\left\| \rm{BA} \right\|$?

Let $\left\| . \right\|$ be a unitarily invariant norm on $M_n$. If $A, B ∈ M_n$, $A$ is normal, and $B$ is Hermitian, why does $\left\| \rm{AB} \right\|{\rm{ = }}\left\| \rm{BA} \right\|$?