54 J.Zhuo Appl Muth Comput.92 (1998,49 58 LE is low triangular and the Cholesky factor of A.L8=B(L) L.ER""is low triangular and the Cholesky factor of LaLg. 3.Generalized semi-positive definite case Consider the linear system (8))() (11 where matrices A,B and C satisfy Condition I. Let 4-B G=(8c (12) and obviously.the matrix G3 is nonsymmetric.But the symmetric part of G3. i.e..(G+G3)/2.is symmetric semi-positive definite [2.5].So we call G3 the generalized semi-positive definite. As in the above discussion,we can always form the low triangular matrix L from symmetric positive definite matrix A and La B(LI) Since C'is symmetric semi-positive definite and La full row rank.there always exists the Cholesky factorization,i.e.. L LI =C+LaLg where L,ER""is low triangular.Thus we have proved the following result. Theorem 3.Assume thut Ga is of where A,B and C satisfv Condition I.Then there always exists the generalized Cholesky factorization. G3 =L3Lg. (13) where =(2) (日) La ER"is low triangular and the Cholesky factor of A.L.ER""is low trian- gulur and the Cholesky factor of C+LaLg.Ln B(LI)E R