# Numerical LA: Cholesky & Conjugate gradient method

#### eckiller

Hello,

I have to do a proof and am having trouble starting.

The proof is to show how you could use Cholesky decomposition to determine a set of A-orthogonal directions.

Cholesky decom. means I can write the symmetric positive definite matrix as

A = GG'

The textbook gives a way of determining the A-orthogonal set using A. Specifically,

v_k = r_k-1 + s_k-1*v_k-1

where v_k is the kth direction vector and r_k-1 is the k-1 residual vector. So we want to choose s_k-1 such that

<v_k-1, Av_k> = 0

The textbook then goes onto show:

s_k-1 = - <v_k-1, Ar_k-1> / <v_k-1, Av_k-1>

So I don't see how using A = GG' helps at all.

If anyone could give me a tip on how to start, I'd be thankful.

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#### fresh_42

Mentor
2018 Award
I assume Cholesky guarantees that you don't go in circles with the given recursions.

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