How can A-orthogonal vector sets be determined using positive definite matrices?

[eckiller]

This is a follow up to a post I made a couple days ago.

Basically, I needed to find a set of a-orthogonal vectors given that A is positive definite.

Is the following satisfactory?

Pick the standard basis B = {e1, ..., en}.

Then consider ei' A ej such that i != j.

Since A is positive definite, A can be factored as A = L'L.

Then (ei' L')(L ej)

However, for all ei and ej s.t. i != j,

(ei' L')(L ej) = 0

ei' A ej = 0

<ei, Aej> = 0

So I have determined an A-orthogonal set.

 

[eckiller]

I made a mistake here:

(ei' L')(L ej) = 0


But I still need help.

 

[mathwonk]

the idea is to project the first vectore onto thes econd and subtract off the projection, so that what remains is ortogonal. so you have to review how to project using a dot products.

 

[eckiller]

Thank you for your reply. However, I am suppose to show this with a Cholesky decomposition. The book shows how to do it w/ the projection method.

I use ' for transpose.

ei is the ith vector of standard basis. i != j

ei' ej = 0

ei' (L inv(L)) ej = 0

... ?

ei' L L' ej = 0

ei' A ej = 0

<ei, Aej> = 0

I'm having trouble filling in "?"

 

[eckiller]

Nevermind, I figured it out...I was trying to prove the wrong thing. Rather, I was trying to prove something that is not true.