Tagged Questions

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

52 views

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 ...
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. ...
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 ...
56 views

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 ...
29 views

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 $$||A||_{op}=\sup\{||Ax||\space |\space x\in\mathbb{R^n}, ||x||=1\}\space\space\text{(operator norm)}$$ ?
19 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 ...
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 ...
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 $u(x,t)=u_0 (x)$ ...
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.
22 views

23 views

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: $$||a x - b y || \leq ... 0answers 46 views Show that ||x|| is a norm on \mathbb{R^n} Let A\subset \mathbb{R^n} be any limited, open, convex, and the centre symmetry set having centre at 0. Show that ||x|| = \inf \{k>0 : x/k \in A \} is a norm at \mathbb{R^n} and open ball ... 1answer 18 views \|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0 Why is A injective? Why does A:X\to Y where X is a banach space and Y is a normed space, where A is a surjective bounded linear operator, where:$$\|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0 Mean that $A$ is also an ...
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 ...