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?
Best Answer
This determines a norm $\|\_\|_P$ iff $P$ is positive definite, then it naturally defines a metric by $d(x,y):=\|y-x\|_P$.
Else the above $d$ function would not be defined or would fail to be a metric, as e.g. there could be $x\ne 0$ with $x^TPx\le0$.
All in all, what it basically says is that the quadratic form $x\mapsto x^TPx$ can be viewed as a generalisation of $x\mapsto \|x\|^2$.
For more details and geometric insights, see for example Pseudo Euclidean spaces.