Diagonally dominant matrix for Cholesky?

Question

I have a $10^6 \times 10^6$ dense SPD matrix, which I am called to invert, by using Cholesky factorization.

However, I came across this statement:

We start with the Cholesky and LU decompositions, which work only with positive definite and nonsingular diagonally dominant square matrices, respectively (Sects. 4.1.1 and 4.1.2).

Source

If I go to the relevant section, I see nothing relevant. I hadn't thought of the diagonally dominant property, should I?

Answer 1

The keyword here is "respectively". Cholesky works with positive definite, and LU works with diagonally dominant.