We start by stating a version of the Perron–Frobenius theorem. Let A be a d × d stochastic matrix, where here we use this to mean that the entries of A are non-negative, and every column sums to 1: Aij ∈ [0, 1] for all i, j, and ∑d i=1Aij = 1 for all j. Thus the columns of A are probability vectors. Such a matrix A describes a weighted random walk on d sites: if the walker is presently at site ...

and Applied Analysis 3 Every λk is a critical level of F subject to S, and it is achieved at some uk ∈ S, that is, F|S ′ uk 0 7 cf. 4, 5 . It follows that λk is an eigenvalue of 1 and uk is the corresponding eigenvector. In general, the sequence {λk}k 1 given by 6 does not exhaust the set of all critical levels of F|S, and thus it might not be the set of all eigenvalues of 1 . An eigenvalue of ...

The largest eigenvalue of a network’s adjacency matrix and its associated principal eigenvector are key elements for determining the topological structure and the properties of dynamical processes mediated by it. We present a physically grounded expression relating the value of the largest eigenvalue of any network to the largest eigenvalue of two network subgraphs, considered as isolated: The ...

The map which takes a square matrix to its tropical eigenvalue-eigenvector pair is piecewise linear. We determine the cones of linearity of this map. They are simplicial but they do not form a fan. Motivated by statistical ranking, we also study the restriction of that cone decomposition to the subspace of skew-symmetric matrices.

A novel integrated approach is developed to design systems for stability and robustness. First, design parameters with large variation bounds are chosen to maintain system stability. Then, a robust eigenvalue design problem is considered to make the dynamic response less sensitive to parameter variations. A new complex sensitivity matrix is derived from the system dynamics with the eigenvalue v...

Graph partitioning problems are NP-complete and various heuristic algorithms exist in the litterature. Particularly, spectral graph partitioning algorithms partition the graph using the eigenvector associated with the second smallest eigenvalue of the \graph Laplacian." Through the use of graph theory we have devoloped preconditioned subspace algorithms for spectral partitioning.

Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ≤ r−1, we show that the proposed method exhibits an error of the order of 4r for eigenvalue approximation and of the order of 3r fo...

The pairwise comparisonsmethod is a convenient tool used when the relative order among different concepts (alternatives) needs to be determined. One popular implementation of the method is based on solving an eigenvalue problem for the pairwise comparisons matrix. In such cases the ranking result the principal eigenvector of the pairwise comparisonmatrix is adopted, whilst the eigenvalue is use...

As the performance of hardware is limited, the focus has been to develop objective, optimized and computationally efficient algorithms for a given task. To this extent, fixed-point and approximate algorithms have been developed and successfully applied in many areas of research. In this paper we propose a feature selection method based on fixed-point algorithm and show its application in the fi...

We present a topology-based segmentation as means for visualizing 2D symmetric tensor fields. The segmentation uses directional as well as eigenvalue characteristics of the underlying field to delineate cells of similar (or dissimilar) behavior in the tensor field. A special feature of the resulting cells is that their shape expresses the tensor behavior inside the cells and thus also can be co...