Is This Norm Equality Correct for \( Ax \)?

  • Thread starter Thread starter ahamdiheme
  • Start date Start date
  • Tags Tags
    Norm
ahamdiheme
Messages
24
Reaction score
0
I just want to verify if the following is correct
\left\right\|x\|2.\left\right\|A\|2= \left\right\|Ax\|2

Thanks
 
Physics news on Phys.org
How are you defining the norm of the operator? As Tr(A^TA)? In that case, no. If you let x=(1,0,...,0), the right-hand side only contains components from the first column of A but the left-hand side contains other components.

However, the norm of A is often defined as

\|A\|=\sup_{\|x\|=1}\|Ax\|

so maybe if you change the = to ≤...
 
Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...
View full post »

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Similar threads

I A true or false statement concerning condition number of a matrix
Replies
2
Views
576
I What is the size of the quotient group L/pZ^m?
Replies
1
Views
2K
Question about induced matrix norm
Replies
14
Views
2K
I Existence and Uniqueness of Inverses
Replies
5
Views
2K
MHB Equivalent Norms: Proving $\|A\| \leq \|A\|_{Eucl} \leq \sqrt{n}\|A\|$
Replies
4
Views
2K
Norm indueced by a matrix with eigenvalues bigger than 1
Replies
8
Views
2K
I Understanding zero divisors & ##\mathbb{Z_m}## in Ring Theory
2
Replies
55
Views
6K
I Understanding the ##ε## as used in limits of sequences
Replies
2
Views
2K
I Proof concerning the Four Fundamental Spaces
Replies
14
Views
2K
I On a bound on the norm of a matrix with a simple pole
Replies
8
Views
3K

Hot Threads

Recent Insights

Back
Top