ABSTRACT
Since Qp is positive definite Qp-1 is also positive definite so that by continuing the procedure of completing the square, we can write
where T is the unique upper triangular matrix
with Tii>0, i=1,…, p. Q.E.D. Thus a symmetric positive definite matrix A can be uniquely written as A = T'T
where T is the unique nonsingular upper triangular matrix with positive diagonal elements. From (1.12) it follows that
so that we can write
Hence, given any symmetric positive definite matrix A there exists a unique nonsingular lower triangular matrix T with positive diagonal elements, such that A=T'T. Let ? be an orthogonal matrix in the diagonal form. For any upper (lower) triangular matrix T, ?T is also an upper (lower) triangular matrix and T'T=(?T)'(?T). Thus given any symmetric positive definite matrix A, there exists a nonsingular lower triangular matrix T, not necessarily with positive diagonal elements, such that A=T'T. Obviously such decomposition is not unique.
