A positive definite matrix A can be factored into
A = T'T, (2.79)
where T is a nonsingular upper triangular matrix. One way to obtain T is the Cholesky decomposition, which can be carried out in the following steps.
Let A = (aij) and T = (tij) be n x n. Then the elements of T are found as follows:
t11 = sqrt(a11), t1j = a1j / t11 2 <= j <= n
tii = sqrt(aii - sum_{k=1}^{i-1} t_{ki}^2) 2 <= i <= n
tij = (aij - sum_{k=1}^{i-1} t_{ki} t_{kj}) / tii 2 <= i < j <= n
tij = 0 1 <= j < i <= n
For example, let
A = (3 0 -3; 0 6 3; -3 3 6).
Then by the Cholesky method, we obtain
T = (sqrt(3) 0 -sqrt(3); 0 sqrt(6) sqrt(1.5); 0 0 sqrt(1.5)),
T'T = (sqrt(3) 0 0; 0 sqrt(6) 0; -sqrt(3) sqrt(1.5) sqrt(1.5))(sqrt(3) 0 -sqrt(3); 0 sqrt(6) sqrt(1.5); 0 0 sqrt(1.5))
= (3 0 -3; 0 6 3; -3 3 6) = A.