Quadratic Forms

Definition

A quadratic form is a multivariate polynomial each term of which has degree two.

Representation

Let $f: \R ^n \to \R $ be a quadratic form. There exists a matrix $A \in \R ^{n \times n}$ so that

\[ x^\top A x = \sum_{i,j} A_{ij}x_{i}x_{j}. \]

Suppose $A$ is not symmetric. Then

\[ f(x) = f(x)/2 + f(x)/2 = \frac{1}{2}x^\top Ax + \frac{1}{2}x^\top A^\top x = x^\top (\frac{1}{2}(A + A^\top ))x. \]

Define $B = 1/2 (A^\top + A)$. We call $B$ the symmetric part of $A$. Since every matrix $A$ has a symmetric part, we can always assume that the matrix for a quadratic form is symmetric. If it is not, replace it with its symmetric part, obtaining the same function.

Under trace

Observe that $\tr x^\top A x = \tr A x x^\top = \tr xx^\top A$.