Why is this expression so common in statistics and machine learning?

I am talking about the expression $x^{T}Ax$. And, this is usually followed up mostly with some relevance to eigen values and eigen vectors. I have seen it in several problems, theoretical discussions and a whole lot of other places. Can anyone give an insightful explanation about this?

• It is common throughout mathematics: it is a quadratic form. Wikipedia states "Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form)."
– whuber
Jul 3 '17 at 15:02
• Don't know if it's a correct way to put it, but is there some geometrical significance that can be inferred generally? Rotation or something else? Jul 3 '17 at 15:34
• To answer that, pursue the references to linear algebra, group theory, differential geometry, differentiable topology, and Lie Theory: they all concern some form of geometry. (So do the number theory applications, for that matter, because they can be understood geometrically, too--it's just not as obvious.)
– whuber
Jul 3 '17 at 15:42