Clustering, multicollinearity, and singular vectors

Let $A$ be a matrix with its pseudo-matrix $A^{†}$ and set $S=I-A^{†}A$. We prove that, after re-ordering the columns of $A$, the matrix $S$ has a block-diagonal form where each block corresponds to a set of linearly dependent columns. This allows us to identify redundant columns in $A$. We explore some applications in supervised and unsupervised learning, specially feature selection, clustering, and sensitivity of solutions of least squares solutions.