What is "quadratic form"

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,

4x + 2xy − 3y
is a quadratic form in the variables x and y.

Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory ( orthogonal group), differential geometry ( Riemannian metric), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form).

In multivariate statistics, if ε is a vector of n random variables, and Λ is an n-dimensional symmetric matrix, then the scalar quantity εΛε is known as a quadratic form in ε.