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

Amira Serrano

Amira Serrano

Answered question

2022-10-15

I have a feeling this is a very easy question but I can't figure out what I'm missing
Let's say we have a quadratic form
f ( x ) = 1 2 x T A x b T x
where 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 x
and 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.

Answer & Explanation

Kristin Myers

Kristin Myers

Beginner2022-10-16Added 12 answers

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 )

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