Quadratic Functions

A quadratic function on $\mathbb{R}^n$ has every term of degree exactly two. That means terms like $x_i^2$ (a variable squared) or $x_i x_j$ (a product of two different variables). The general quadratic function can be written using a matrix: $f(\mathbf{x}) = \mathbf{x}^\top A \mathbf{x}$ where $A$ is an $n \times n$ matrix.

Expanding the matrix product: $\mathbf{x}^\top A \mathbf{x} = \sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j$. The diagonal terms $a_{ii} x_i^2$ contribute the squared terms, while the off-diagonal terms $a_{ij} x_i x_j$ (for $i \neq j$) contribute the cross terms. For example, if $n=2$: $\mathbf{x}^\top A \mathbf{x} = a_{11}x_1^2 + a_{12}x_1 x_2 + a_{21}x_2 x_1 + a_{22}x_2^2.$

A general second-degree function includes a quadratic part, a linear part, and a constant: $f(\mathbf{x}) = \mathbf{x}^\top A \mathbf{x} + \mathbf{b}^\top \mathbf{x} + c$. This is the most general polynomial of degree $\leq 2$ on $\mathbb{R}^n$.