In the book A Linear Systems Primer (by Antsaklis and others), they first mention squared distance of a point x from the origin:

> 
> $$x^{T}x = ||{x}||^2$$ which represents the square of the Euclidean
>  distance of the state from the equilibrium $x=0$.
> 
> 
> 

So far, so good this is basic linear algebra. Then they go on to say:

> 
> In the following discussion, we will employ as a "generalized distance
>  function" the quadratic form given by $${x}^TPx , P={P}^T $$ where $P$
>  is a real $n\times n$ matrix.
> 
> 
> 

I am familiar with this definition of a quadratic form from linear algebra : we interpose a symmetric matrix to weight the variables in different ways. 

But I am not familiar with this as a distance function. Is that mainly to say that it satisfies the requirements of a metric space? Is there a geometric (intuitive) discussion of the sense in which the quadratic form generalizes the more basic notion of a distance?