Hi!,
i'm having trouble to prove that is a vector norm if and only if A is positive definite, in the property that says .
Thanks in advance.
If A is not positive definite then need not be positive, so will not be positive-valued and hence will not be a norm.
If A is positive definite then it has a positive square root, say for some positive definite matrix S. Then (where means the euclidean norm of the vector Sx), and it is easy to check that this satisfies the properties for a norm.