The PCA, or Principal Component Analysis is used to find a lower dimensional subspace that best represents the data, placing the basis of the new linear subspace in the directions that the data varies most.
The PCA diagonalizes the maximum likelihood estimate of the covariance matrix
$ C=\frac{1}{n} \sum_{i=1}^{n} \vec{x_i}\vec{x_i}^T $
by solving the eigenvalue equation
$ C\vec{e} = \lambda \vec{e} $
The solutions to these equations are eigenvalues $ \lambda_1 \lambda_2 \cdots \lambda_m $. Often only $ k m $ eigenvalues will have a nonzero value, meaning that the inherent dimensionality of the data is $ k $, being $ n-k $ dimensions noise in the data.
In order to represent the data in the k dimensional space we first construct the matrix $ E=[\vec{e_1} \vec{e_2} \cdots \vec{e_k}] $. The projection to the new k-dimensional subspace is done by the following linear transformation:
$ \vec{x}^{'} = E^T\vec{x} $
