Numerical results on noisy blown-up matrices

Abstract

We study the eigenvalues of large perturbed matrices. We consider an Hermitian pattern matrix 𝑃 of rank 𝑘 . We blow up 𝑃 to get a large block-matrix 𝐵 𝑛 . Then we generate a random noise 𝑊 𝑛 and add it to the blown up matrix to obtain the perturbed matrix 𝐴 𝑛 = 𝐵 𝑛 + 𝑊 𝑛 . Our aim is to find the eigenvalues of 𝐵 𝑛 . We obtain that under certain conditions 𝐴 𝑛 has 𝑘 ‘large’ eigenvalues which are called structural eigenvalues. These structural eigenvalues of 𝐴 𝑛 approximate the non-zero eigenvalues of 𝐵 𝑛 . We study a graphical method to distinguish the structural and the non-structural eigenvalues. We obtain similar results for the singular values of non-symmetric matrices.