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    Proof: Let X* minimize the quadratic function Q(X). Then

                  Q(X*)=B+A X*=0                        (6.32)

    Given a point X  and a set of linearly independent directions S ,S ,…,S ,constants  can always be found such that

                   X*= X +                            (6.33)

    where the vectors S ,S ,…,S  have been used as basis vectors. If the directions S are A conjugate and none of them is zero, the S , can easily be shown to be linearly independent and the  can be determined as follows.

    Equations (6.32) and (6.33) lead to

               B+A X +A( )=0                       (6.34)

       Multiplying this equation throughout by S ,we obtain

                  S  (B+A X )+ S A( )=0               (6.35)

       Equation (6.35) can be rewritten as

                  (B+A X ) S + S A S =0                  (6.36)

    that is, 

                   =-                           (6.37)

    Now consider an iterative minimization procedure starting at point X ,and successively minimizing the quadratic Q(X) in the directions S ,S ,…,S ,where these directions satisfy Eq. (6.27). The successive points are determined by the relation

                  X =X + S ,  i=1 to n                     (6.38)

    where  is found by minimizing Q (X + S )so that S  Q(X )=0 is equivalent to  =0 atY= X 

                          = 

    where y  are the components of Y= X  

                  S  Q(X )=0                              (6.39)

    Since the gradient of Q at the point X , is given by 

                   Q(X )=B+A X                           (6.40)

    Eq. (6.39) can be written as

                  S {B+A(X + S )}=0                        (6.41)

    This equation gives

                  =-                              (6.42)

    From Eq. (6.38), we can express X  as

                 X =X +                                 (6.43)

    so that

                       X AS = X AS + 

                         = X AS                               (6.44)

    using the relation (6.27). Thus Eq. (6.42) becomes

                  =-(B+AX )                           (6.45)

    which can be seen to be identical to Eq. (6.37). Hence the minimizing step lengths are given by   or   . Since the optimal point X* is originally expressed as a sum of n quantities   which have been shown to be equivalent to the minimizing step lengths, the minimization process leads to the minimum point in n steps or less. Since we have not made any assumption regarding X  and the order of S ,S ,…,S , the process converges in n steps or less, independent of the starting point as well as the order in which the minimization directions are used.

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