principal components - change of basis

Change of Basis Covariance

Starting with a random vector {% \vec{Z} %}, we define the mean vector {% \vec{m} %} to be

{% \vec{m} = \mathbb{E}(\vec{Z}) %}

The the covariance of {% A\vec{Z} %}, where {% A %} is a given transformation matrix is

{% cov(\vec{AZ}) = \mathbb{E}[(A\vec{Z} - A\vec{m})(A\vec{Z} - A\vec{m})^T] %}

{% = \mathbb{E}[A(\vec{Z}-\vec{m})(\vec{Z}-\vec{m})^T A^T] %}

{% = A\mathbb{E}[(\vec{Z}-\vec{m})(\vec{Z}-\vec{m})^T]A^T %}

{% = A cov(\vec{Z}) A^T %}