Assume that we have -dimensional data vectors , which form the data matrix = . The matrix is decomposed into
Without any further constraints, there exist infinitely many ways to perform such a decomposition. However, the subspace spanned by the column vectors of the matrix , called the principal subspace, is unique. In PCA, these vectors are mutually orthogonal and have unit length. Further, for each , the first vectors form the -dimensional principal subspace. This makes the solution practically unique, see [4,2,5] for details.
There are many ways to determine the principal subspace and components [6,4,2]. We will discuss three common methods that can be adapted for the case of missing values.