Overview
Rayleigh Quotient
The Rayleigh Quotient is defined to be
{% \frac{\vec{x}^T A \vec{x}}{\vec{x}^T \vec{x}} %}
Given a matrix {% A %} with the following eigenvalue/eigenvector pair
{% A\vec{x} = \lambda \vec{x} %}
The quotient simplies to
{% \frac{\vec{x}^T A \vec{x}}{\vec{x}^T \vec{x}} = \lambda \frac{\vec{x}^T \vec{x}}{\vec{x}^T \vec{x}} = \lambda %}
The eigenvector with the largest eigenvalue solves the problem of optimizing the Rayleigh Quotient.
(see Shawe-Taylor)