Overview
Given a covariance matrix {% \Sigma %} that is symmetric and positive definite, there exists a set of vectors {% \vec{v}_1, \vec{v}_2 , ..., \vec{v}_m %} such that
{% \Sigma \vec{v}_i = \alpha_i \vec{v}_i %}
and
{% \langle \vec{v}_i| \vec{v}_j \rangle = \delta_{i,j} %}
(that is, the vectors are orthonormal)
Note: here we are using the
Dirac Notation