How to Use Cholesky Factorization for Positive Definite Matrices

[freshnitesh]

Homework Statement

2. The attempt at a solution

So part a. makes sense to me, it basically comes down to

A1 =
1 -1 -1
-1 1 1
-1 1 1

A2 =
1 -1 -1
-1 2 -2
-1 -2 11

I'm not sure how to approach part b. because the question doesn't make much sense to me. It's asking to show that f1 is a single perfect square, so my guess is that from A1 I have to derive an expression for f1 that is a single perfect square.

Haven't really looked to hard at part c, yet.

 

[Ray Vickson]

Homework Statement

2. The attempt at a solution

So part a. makes sense to me, it basically comes down to

A1 =
1 -1 -1
-1 1 1
-1 1 1

A2 =
1 -1 -1
-1 2 -2
-1 -2 11

I'm not sure how to approach part b. because the question doesn't make much sense to me. It's asking to show that f1 is a single perfect square, so my guess is that from A1 I have to derive an expression for f1 that is a single perfect square.

Haven't really looked to hard at part c, yet.

Use Chpolesky factorization, which works for any positive-semidefinite matrix. (Many discussions assume positive definiteness, but if you go through the material carefully you can show that it applies as well to the semidefinite case.) See, eg., http://en.wikipedia.org/wiki/Cholesky_decomposition .
In fact, using the Cholesky algorithm is by far the easiest way to test positive definitness or semidefinitness---much easier than dealing with determinants, for example.