MHB How to Solve Condition Number and LU Decomposition Problems?

[akerman]

I have two question one of them I have solved but a bit differently and the second is something I need more help with. View attachment 2466

First question I have solved previously but bit different and I am not too sure how it should be solved in part b given above. Here is my similar solution View attachment 2467

Can you comment and show what should be done for part b and also for part c which I didn't know how to show.

P.S. I am preparing for my exams and this is not a coursework or anything in terms of homework. Therefore, explanation and comments what could be improved are good for me. Thanks

 

[I like Serena]

Hi akerman!

There is no mention in problem statement (b) of $\widetilde A$ or $\delta A$.
So it seems to me it should not be involved...

For (c), I would suggest to write:
$$a_{i,j} \overset ?= \sum_k l_{i,k} u_{k,j}$$
And write it out knowing that e.g. $l_{i,k} = 0$ unless $k=i$ or $k=i+1$.

 

[akerman]

Hi akerman!

There is no mention in problem statement (b) of $\widetilde A$ or $\delta A$.
So it seems to me it should not be involved...

For (c), I would suggest to write:
$$a_{i,j} \overset ?= \sum_k l_{i,k} u_{k,j}$$
And write it out knowing that e.g. $l_{i,k} = 0$ unless $k=i$ or $k=i+1$.

For (c) would it be enough to show the proof that product of two lower triangular matrices is still lower triangular and the same thing for upper triangular?

 

[I like Serena]

For (c) would it be enough to show the proof that product of two lower triangular matrices is still lower triangular and the same thing for upper triangular?

Those matrices are not just lower respectively upper triangular.
They are band matrices with specific values in the bands.
Any proof should take that into account and show that those specific values will match.