Positive definite symmetric 配列

I mentioned positive definite symmetric matrices in conjunction with LU factorization via systolic arrays in the previous post.
A symmetric matrix is one which equals its transpose, e.g.
1 2 3
2 9 4
3 4 7
A positive definite matrix is one in which (z**T)Mz>0 for all non-zero vectors z with real entries. z**T denotes the transpose of z (write in row format).
I am unsure as to how this enables LU factorization to be carried out via Gaussian elimination without pivoting. Nor am I sure how the recurrences cited in the paper for elements of matrices A, L, U are derived.