be a set of N column vectors of dimension D.
Define the scatter matrix 
of the data set as
where
is the mean of the dataset
Bob Fisher
Principal component analysis can be used to analyze the structure of a data set or allow the representation of the data in a lower dimensional dataset (as well as many other applications).
Let be a set of N column vectors of dimension D.
Define the scatter matrix of the data set as
where is the mean of the dataset
The d largest principle components are the eigenvectors
corresponding to the d largest eigenvalues.
d can be chosen arbitrarily with d < D.
The eigenvectors of S can usually be found by using
singular value decomposition.
The dominant eigenvectors describe the main directions of variation of the data. For example, if a dataset had 2 large eigenvalues, then the data variation is described largely by linear combinations of the 2 corresponding eigenvectors (ie. the data is largely coplanar).
The d eigenvectors can also be used to project the data into a
d dimensional space.
Define
The projection of vector 
is 
.
The corresponding scatter matrix 
of the
vectors 
is:
The matrix W maximizes the determinant of
for a given d.