Find Cholesky Decomposition for 3x3 Matrix | Linear Algebra Question"

[stunner5000pt]

ok question is to find the Cholesky Decomposition of this matrix

$$\left(\begin{array}{ccc}2&-1&0\\-1&2&-1\\0&-1&2\end{array} \right)$$

now for the cholesky decomposition L i know how to find the first column that is
$$\left(\begin{array}{ccc}\sqrt{2}&0&0\\-\frac{1}{\sqrt{2}}&\frac{3}{2}&?\\0&?&?\end{array}\right)$$

the qeustion marks mean that parts i can't figure out

can someone please lhelp me !

thank you

 

[CarlB]

Gosh, it seems like you're on the right direction. Why don't you put A, B, and C into those question marks, take the transpose, multiply the two matrices and see what equations you get on A, B and C?

Carl

 

[stunner5000pt]

isn't there a more direct approach that doesn't involve matrix multiplication?

like how i got the 3/ in the a22 position was as follows

$$\sqrt{2 - \frac{1}{2}}$$
but how would ig oabout findin the ? in the a32 postiion??

is it $$\frac{-1-0}{\frac{3}{2}}$$
is that correct??

 

[CarlB]

You can use an algorithm like:
http://en.wikipedia.org/wiki/Choleski_factorization

But it seems to me that that's the hard way.

Carl

By the way, thanks for bringing this to my attention. I think it's a very useful thing to know.

 

[stunner5000pt]

i don't want to do it the wikipedia way its too hard i think. How would you do this question??

PLease i need to figure this out !

 

[stunner5000pt]

any help... anyone? i know that I am not suppsoed to bump liek this but how would one go about doing this decomposition?

For that matter, how would do the LDL transpose factorization for a matrix

one method suggested to me was to reduce the origina lamtrix to the elementary matrix, and apply each of the row operatorions to a differnet elementary matrix and then finally mulitply them all so give the Lower triangular matrix

but how would one go about fdind the D matrix?

 

[CarlB]

The matrix multiplication for this problem should be fairly easy. That's what I would do.

You're only going to end up with three equations in three unknowns. They will be quadratic in the unknowns, but you should be able to solve them without a lot of trouble.

Carl

Oh, and by the way, my Seahawks just one in overtime, improving their record to 9-2-0 continuing a win streak now at 7 games.