A matrix A is positive definite if and only if the determinants of all of its leading minors are positive. Use the formula of the determinant of a partitioned matrix to show that det(A[k] ) = L2 kk...

A matrix A is positive definite if and only if the determinants of all of its leading minors are positive. Use the formula of the determinant of a partitioned matrix to show that det(A[k] ) = L2 kk det(A[k−1]), so that Cholesky factorization will work if and only if the matrix is positive definite.