Quadratic forms in random variables

Proof. The “if” part is proven by construction. The Cholesky decomposition, R, is constructed a row at a time and the diagonal elements are evaluated as the square roots of expressions calculated from the current row of A and previous rows of R. If the expression whose square root is to be calculated is not positive then you can determine a non-zero x ∈ Rk for which x′Ax ≤ 0. Suppose that A = R′R with R invertible. Then x′Ax = x′R′Rx = ‖Rx‖ ≥ 0 with equality only if Rx = 0. But if R−1 exists then x = R−10 must also be zero.