How to Estimate the Operator Norm ||A||_2 for a Difference Operator?

[Max Fleiss]

Greetings everyone!

I have a set of tasks I need to solve using using operator norms, inner product... and have some problems with the task in the attachment. I would really appreciate your help and advice.

This is what I have been thinking about so far:
I have to calculate a non trivial upper bound, so maybe it could be done by:
$$b=max( ||A||_1,||A||_2,||A||_\infty )$$

Since $$A$$ is a difference operator I estimated the following:
$$||A||_1= 4$$
$$||A||_\infty= 2$$
But how can I estimate $$||A||_2=?$$
If I know that abs row sum is 2 (besides 0 there appears only one 1 and one -1 in the rows) and abs column sum is 4 (it is two times the size of row length dim(A)=2mn x mn). Can I estimate $$||A||_2$$ by:
$$||A||_2=\sqrt{rows^2+columns^2 }=\sqrt{(2 \cdot 2mn)^2+(4 \cdot mn)^2}$$$$=4 \sqrt{(mn)^2+(mn)^2}=4 \sqrt{2} \sqrt{m^2n^2}$$ since $$mn$$ are positive $$||A||_2=4 \sqrt{2} mn$$
So I would say $$b=max(L_1,L_2,L_\infty)=L_2=4 \sqrt{2} mn$$

Is my conclusion, approximation of a non trivial upper bound b right?

Thank you in advance for your help!

 

[fresh_42]

It is not clear why you use different norms here. In any case, you need to investigate $||Ax||$ for any $x$. Application of $A$ on $x$ should give you an idea for un upper bound, i.e. you need to use the definition of $A$ somewhere, since not all operators are bounded.

 

[math771]

Hi there,

Thanks for sharing your thoughts on this problem. It seems like you have made some good progress so far. However, I have a few suggestions for your approach.

Firstly, it is important to note that the operator norm ||A||_2 is defined as the maximum singular value of A, which is different from your estimation of ||A||_2 as the square root of the sum of squares of the row and column sums. So, your estimation may not be accurate.

To calculate the operator norm, you can use the singular value decomposition (SVD) of A, which gives you the singular values of A in descending order. The maximum singular value will give you the operator norm.

Also, it is not necessary to use the max function to find the upper bound b. You can simply use the operator norms ||A||_1, ||A||_2, and ||A||_\infty to find an upper bound.

I would also recommend double-checking your estimations for ||A||_1 and ||A||_\infty, as they seem a bit low for a difference operator.

I hope this helps. Good luck with your task!