Perron Complement Research Articles

Perron complement and Perron root

For a nonnegative irreducible matrix A, this paper is concerned with the estimation and determination of the unique Perron root or spectral radius of A. We present a new method that utilizes the relation between Perron roots of the nonnegative matrix and its (generalized) Perron complement. Several numerical examples are given to show that our method is effective, at least, for some classes of nonnegative matrices.

A divide and conquer approach to computing the mean first passage matrix for Markov chains via Perron complement reductions

AbstractLet MT be the mean first passage matrix for an n‐state ergodic Markov chain with a transition matrix T. We partition T as a 2×2 block matrix and show how to reconstruct MT efficiently by using the blocks of T and the mean first passage matrices associated with the non‐overlapping Perron complements of T. We present a schematic diagram showing how this method for computing MT can be implemented in parallel. We analyse the asymptotic number of multiplication operations necessary to compute MT by our method and show that, for large size problems, the number of multiplications is reduced by about 1/8, even if the algorithm is implemented in serial. We present five examples of moderate sizes (of orders 20–200) and give the reduction in the total number of flops (as opposed to multiplications) in the computation of MT. The examples show that when the diagonal blocks in the partitioning of T are of equal size, the reduction in the number of flops can be much better than 1/8. Copyright © 2001 John Wiley & Sons, Ltd.

On Perron complements of totally nonnegative matrices

An n×n matrix is called totally nonnegative if every minor of A is nonnegative. The problem of interest is to describe the Perron complement of a principal submatrix of an irreducible totally nonnegative matrix. We show that the Perron complement of a totally nonnegative matrix is totally nonnegative only if the complementary index set is based on consecutive indices. We also demonstrate a quotient formula for Perron complements analogous to the so-called quotient formula for Schur complements, and verify an ordering between the Perron complement and Schur complement of totally nonnegative matrices, when the Perron complement is totally nonnegative.

Inverses of Perron complements of inverse M-matrices

The concept of the Perron complement of a nonnegative and irreducible matrix was introduced by Meyer in 1989 and it was used by him to construct an algorithm for computing the stationary distribution vector for Markov chains. Here we consider properties of the Perron complement of an n×n matrix K which is an inverse of an irreducible M-matrix. We first show that the Perron complements of K are inverses of M-matrices and that the inverses of associated principal submatrices of K are sandwiched between the inverses of the Perron complements of K and the inverses of the corresponding Schur complements of K. We then investigate the directed graph of the inverse of the Perron complements of such matrices K.