Quadratic Forms

Definition

A quadratic form in $n$ variables is a function of the form:

$$f(x) = \vec{x}^T A \vec{x}$$

Here, $A$ is a symmetric $n \times n$ matrix and $x$ is in $R^n$. $A$ is called the matrix associated with $f$.

Example:

$$f(x) = \vec{x}^T A \vec{x}\\ = \begin{bmatrix}x_1&x_2\end{bmatrix} \begin{bmatrix}2&-3\\-3&5\end{bmatrix} \begin{bmatrix}x_1\\x_2\end{bmatrix}\\ = 2x_1^2 + 5x_2^2 - 6x_1 x_2$$

Diagonalization

By using its associated matrix you can diagonalize a quadratic form like any other symmetric matrix. This provides insight into what the function does exactly (like any other diagonalization). The change of variables will also speed up computation if you're doing a lot of consecutive transformations with it, because there will not be any cross-product terms; you can just raise the diagonal matrix to $n$th power for $n$ transformations.

Expressing the diagonalization as an algebraic formula works as follows:

1. Orthogonally diagonalize associated, symmetric matrix $A$ as $A = Q D Q^T$.

2. $x = Qy$, or similarly, $y = Q^T x$.

For a great example, see page 412 of David Poole's book on Linear Algebra.