Direct formulation to Cholesky decomposition of a general nonsingular correlation matrix

We present two novel, explicit representations of Cholesky factor of a nonsingular correlation matrix. The first representation uses semi-partial correlation coefficients as its entries. The second, uses an equivalent form of the square roots of the differences between two ratios of successive determinants. Each of the two new forms enjoys parsimony of notations and offers a simpler alternative to both spherical factorization and the multiplicative partial correlation Cholesky matrix (Cooke et al 2011). Two relevant applications are offered for each form: a simple $t$-test for assessing the independence of a single variable in a multivariate normal structure, and a straightforward algorithm for generating random positive-definite correlation matrix. The second representation is also extended to any nonsingular hermitian matrix.