I have a feeling this is a very easy question but I can't figure out what I'm missingLet's say we have a quadratic form f ( x ) = 1 2 x T A x − b T xwhere A is a n × n symmetric and positive definite matrix and x,b are n × 1 vectors.If we introduce the weighted norm as ‖ x ‖ A 2 = x T A xand x∗ is the solution of the system Ax=b, then I want to prove that 1 2 ‖ x − x ∗ ‖ A 2 = f ( x ) − f ( x ∗ )What I've done so far: 1 2 ‖ x − x ∗ ‖ A 2 = 1 2 ( x − x ∗ ) T A ( x − x ∗ ) = 1 2 ( x T A x − x T A x ∗ − x ∗ T A x + x ∗ T A x ∗ ) = 1 2 ( x T A x − b T x − b T x + x ∗ T A x ∗ ) = 1 2 x T A x − b T x + 1 2 x ∗ T A x = f ( x ) + 1 2 x ∗ T A x ∗ I'm missing a term − b T x ∗ to complete the proof but no matter how I look at it I can't find where I've made a mistake, if there is one.

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I have a feeling this is a very easy question but I can&#039;t figure out what I&#039;m missingLet&#039;s say we have a quadratic form  f  (  x  )  =      1    2        x    T    A  x  −      b    T    xwhere A is a   n  ×  n symmetric and positive definite matrix and x,b are   n  ×  1 vectors.If we introduce the weighted norm as  ‖  x      ‖    A    2    =      x    T    A  xand x∗ is the solution of the system Ax=b, then I want to prove that      1    2    ‖  x  −      x    ∗        ‖    A    2    =  f  (  x  )  −  f  (      x    ∗    )What I&#039;ve done so far:      1    2    ‖  x  −      x    ∗        ‖    A    2      =      1    2    (  x  −      x    ∗        )    T    A  (  x  −      x    ∗    )    =      1    2    (      x    T    A  x  −      x    T    A      x    ∗    −      x          ∗      T        A  x  +      x          ∗      T        A      x    ∗    )    =      1    2    (      x    T    A  x  −      b    T    x  −      b    T    x  +      x          ∗      T        A      x    ∗    )    =      1    2        x    T    A  x  −      b    T    x  +      1    2        x          ∗      T        A  x  =    f  (  x  )  +      1    2        x          ∗      T        A      x    ∗  I&#039;m missing a term   −      b    T        x    ∗   to complete the proof but no matter how I look at it I can&#039;t find where I&#039;ve made a mistake, if there is one.

Kristin Myers · Accepted Answer

1    2        x          ∗      T        A      x    ∗    =  −  (      1    2        x          ∗      T        A      x    ∗    −      x          ∗      T        A      x    ∗    )  =  −  (      1    2        x          ∗      T        A      x    ∗    −      x          ∗      T        b  )  =  −  (      1    2        x          ∗      T        A      x    ∗    −      b          T            x    ∗    )  =  −  f  (      x    ∗    )

Prove that (1)/(2) norm(x−x)_A^2=f(x)−f(x)

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