Some Results and Connections of an Eigendecomposition Problem

TL;DR

This paper investigates the eigendecomposition of matrices derived from special incidence matrices, revealing orthogonal eigenvectors and formulas for related matrices, with applications in computational biology.

Contribution

It provides a concrete description of eigenvectors for a class of incidence matrices and derives formulas for associated matrices, connecting algebraic properties with combinatorial structures.

Findings

Eigenvectors are pairwise orthogonal and satisfy specific properties.

Formulas for entries of matrices W and WA are obtained.

Applications in computational biology are discussed.

Abstract

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. .