Cholesky decomposition

As I'm sure you've figured out by now, there is more than one way to factorize a matrix, and there are special methods for special matrices.

The Cholesky decomposition is square root-like and works only on symmetric positive definite matrices. 

This works by factorizing A into the form LL^T. Here, L, as before, is a lower triangular matrix.

Do develop some intuition. It looks like this:

However, here, L is called a Cholesky factor. 

Let's take a look at the case where .

We know from the preceding matrix that ; therefore, we have the following:

Let's multiply the upper and lower triangular matrices on the right, as follows:

Writing out A fully and equating it to our preceding matrix gives us the following:

We can then compare, element-wise, the corresponding entries of A and LL^T and solve algebraically for _, as follows:

We can repeat this process for any symmetric positive definite matrix, and compute the l_i,j values given a_i,j.