How to deduce a proper eigenvalue cluster from a proper singular value cluster in the nonnormal case

We consider a generic sequence of matrices (the nonnormal case is of interest) showing a proper cluster at zero in the sense of the singular values. By a direct use of the notion of majorizations, we show that the uniform spectral boundedness is sufficient for the proper clustering at zero of the eigenvalues: if the assumption of boundedness is removed, then we can construct sequences of matrices with a proper singular value clustering and having all the eigenvalues of an arbitrarily big modulus. Applications to the preconditioning theory are discussed

A preconditioning approach to the pagerank computation problem

AbstractSome spectral properties of the transition matrix of an oriented graph indicate the preconditioning of Euler–Richardson (ER) iterative scheme as a good way to compute efficiently the vertexrank vector associated with such graph. We choose the preconditioner from an algebra U of matrices, thereby obtaining an ERU method, and we observe that ERU can outperform ER in terms of rate of convergence. The proposed preconditioner can be updated at a very low cost whenever the graph changes, as is the case when it represents a generic set of information. The particular U utilized requires a surplus of operations per step and memory allocations, which make ERU superior to ER for not too wide graphs. However, the observed high improvement in convergence rate obtained by preconditioning and the general theory developed, are a reason for investigating different choices of U, more efficient for huge graphs