We give a coherent theory of root polynomials, an algebraic tool useful for the analysis of matrix polynomials. In particular, we first survey results previously appeared in the literature, giving a formal proof for those that lacked one. We next extend some of these results providing some new concepts and related theorems, thus simplifying and expanding the theory. Then, we give some applications of root polynomials, such as the recovery of Jordan chains from linearizations of matrix polynomials, or the behaviour of Jordan chains under rational reparametrization. We also briefly discuss how root polynomials can be used to define eigenvectors and eigenspaces for singular matrix polynomials.

The theory of zeta functions provides an expression for the generating function of nonbacktracking walk counts on a directed network. We show how this expression can be used to produce a centrality measure that eliminates backtracking walks at no cost. We also show that the radius of convergence of the generating function is related to the spectrum of a three-by-three block matrix involving the original adjacency matrix. This gives a means to choose appropriate values of the attenuation parameter. We find that three important additional benefits arise when we use this technique to eliminate traversals around the network that are unlikely to be of relevance. First, we obtain a larger range of choices for the attenuation parameter. Second, a natural approach for determining a suitable parameter range is invariant under the removal of certain types of nodes, we can gain computational efficiencies through reducing the dimension of the resulting eigenvalue problem. Third, the dimension of the linear system defining the centrality measures may be reduced in the same manner. We show that the new centrality measure may be interpreted as standard Katz on a modified network, where self loops are added, and where nonreciprocal edges are augmented with negative weights. We also give a multilayer interpretation, where negatively weighted walks between layers compensate for backtracking walks on the only non-empty layer. Studying the limit as the attenuation parameter approaches its upper bound also allows us to propose a generalization of eigenvector-based nonbacktracking centrality measure to this directed network setting. In this context, we find that the two-by-two block matrix arising in previous studies focused on undirected networks must be extended to a new three-by-three block structure to allow for directed edges. We illustrate the centrality measure on a synthetic network, where it is shown to eliminate a localization effect present in standard Katz centrality. Finally, we give results for real networks.

We introduce and study a new network centrality measure based on the concept of nonbacktracking walks; that is, walks not containing subsequences of the form uvu where u and v are any distinct connected vertices of the underlying graph. We argue that this feature can yield more meaningful rankings than traditional walk-based centrality measures. We show that the resulting Katz-style centrality measure may be computed via the so-called deformed graph Laplacian-a quadratic matrix polynomial that can be associated with any graph. By proving a range of new results about this matrix polynomial, we gain insights into the behavior of the algorithm with respect to its Katz-like parameter. The results also inform implementation issues. In particular we show that, in an appropriate limit, the new measure coincides with the nonbacktracking version of eigenvector centrality introduced by Martin, Zhang and Newman in 2014. Rigorous analysis on star and star-like networks illustrates the benefits of the new approach, and further results are given on real networks.

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by Pellet [Bull. Sci. Math. (2), 5 (1881), pp. 393--395], some results from Bini [Numer. Algorithms, 13 (1996), pp. 179--200] based on the Newton polygon technique, and recent results from Gaubert and Sharify (see, in particular, [Tropical scaling of polynomial matrices, Lecture Notes in Control and Inform. Sci. 389, Springer, Berlin, 2009, pp. 291--303] and [Sharify, Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues, Ph.D. thesis, École Polytechnique, Paris, 2011]). These extensions are applied to determine effective initial approximations for the numerical computation of the eigenvalues of matrix polynomials by means of simultaneous iterations, like the Ehrlich--Aberth method. Numerical experiments that show the computational advantage of these results are presented.\ud \ud \ud Read More: http://epubs.siam.org/doi/abs/10.1137/12088674

We revisit the landmark paper V. Mehrmann, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971-1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key properties of linearizations for matrix polynomials. We also show that every pencil in the double ansatz space is intrinsically connected to a Bézout matrix, which we use to prove the eigenvalue exclusion theorem. In addition our exposition allows for any polynomial basis and for any field. The new viewpoint also leads to new results. We generalize the double ansatz space by exploiting its algebraic interpretation as a space of Bézout pencils to derive new linearizations with potential applications in the theory of structured matrix polynomials. Moreover, we analyze the conditioning of double ansatz space linearizations in the important practical case of a Chebyshev basis.