Some Results and Connections of an Eigendecomposition Problem

We consider the problem of finding nonzero eigenvalues and the corresponding eigenvectors of a matrix $AA^{\top}$, where $A$ is a special incidence matrix; This matrix can equivalently be defined based on a match relation between some sequences. By using a concrete description of the obtained eigenvectors, we show that these are pairwise orthogonal and satisfy nice properties. The combinatorial arguments, in the sequel, lead us to obtain formulas for entries of matrices $W$ and $WA$, where $W$ is the Moore-Penrose pseudo-inverse of $A$. A special case of this problem has previously found applications in computational biology. .