W.Wang,J.Zhao Linear Algebra and its Applications 291 (1999)185-199 187 Theorem 2.2.Let B be an (n +1)x n matrix B }1 in. (3) where D is nonsingular,and M=diag(,...)with >0.Then there is an M-invariant reflection O such that e=(8) (4) Proof.Let c=(e)a=()h We will construct =I-2cdT such that Eq.(4)is satisfied.Then we obtain the relation () 2 cidib+cid D ed b+ed D =(8) (5) giving DTd ub where =(1-2cid)/2c1. By Lemma 2.1,we have c=Md/d Md and c=ud/d Md that inserted in the expression for u gives 2p1d+21d-dMd=0. (6) By expanding dMd and using d=uD-Tb we have d"Md=diu+u(D-Tb)'M(D-Tb) enabling us to rewrite Eg.(6)as 41d+241d-u(DTb)M(D-Tb)=0. Solving this equation for di we get . By choosing u.we obtain d and =I-2MddT/dMd. In order to avoid rounding errors we choose the negative sign of the square root in di. We may choose u =1/lbll2 and if =0,we set I. We have the follwing algorithm for determining the M-invariant reflection