The Spectrum of a Certain Large Block Matrix

Large matrices appear in many applications in computer science, physics, chemistry and many other disciplines. This is because such matrices have the ability to hold huge amounts of memory. On of the main properties that researchers are interested is studying the spectral theory of these matrices. In this paper, we compute the spectrum of a certain large matrix that can serve as an adjacency matrix of a certain clean graph. In particular, we give a full characterization of the eigenvalues and eigenvectors of the intended matrix.