We present an implementation of the Normalized Cuts method for the solution of the image segmentation problem on polygonal grids. We show that in the presence of rounding errors the eigenvector corresponding to the k-th smallest eigenvalue of the generalized graph Laplacian is likely to contain more than k nodal domains. It follows that the Fiedler vector alone is not always suitable for graph ...

Crisp comparison matrices produce crisp weight estimates. It is logical for an interval or fuzzy comparison matrix to give an interval or fuzzy weight estimate. In this paper, an eigenvector method (EM) is proposed to generate interval or fuzzy weight estimate from an interval or fuzzy comparison matrix, which differs from Csutora and Buckley’s LambdaMax method in several aspects. First, the pr...

Partitioning of circuit netlists is important in many phases of VLSI design, ranging from layout to testing and hardware simulation. The ratio cut objective function [29] has received much attention since it naturally captures both mincut and equipartition, the two traditional goals of partitioning. In this paper, we show that the second smallest eigenvalue of a matrix derived from the netlist ...

Let G be a connected graph whose least eigenvalue λ(G) is minimal among the connected graphs of prescribed order and size. We show first that either G is complete or λ(G) is a simple eigenvalue. In the latter case, the sign pattern of a corresponding eigenvector determines a partition of the vertex set, and we study the structure of G in terms of this partition. We find that G is either biparti...

In this paper, we propose an inexact inverse iteration method for the computation of the eigenvalue with the smallest modulus and its associated eigenvector for a large sparse matrix. The linear systems of the traditional inverse iteration are solved with accuracy that depends on the eigenvalue with the second smallest modulus and iteration numbers. We prove that this approach preserves the lin...

The present paper is concerned with the existence2 of the eigenvalue ir of the Hilbert matrix A =((i+k —1)_1) or A = ((i+k)'1), i, k = l, 2, 3, • • • . It is well known that,3 considered as a linear operator in the Hilbert space I2 of vectors with finite square sum of components, A is symmetric, positive-definite and bounded, the upper bound being equal to ir. It is further known that4 ir is no...

1 Linear algebra 2 1.1 Inner product, norm, distance, and orthogonality . . . . . . . . . 2 1.2 Angle and inequality . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Vector projection . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Basics of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . ....

It is known that for a totally positive (TP) matrix, the eigenvalues are positive and distinct and the eigenvector associated with the smallest eigenvalue is totally nonzero and has an alternating sign pattern. Here, a certain weakening of the TP hypothesis is shown to yield a similar conclusion.

We give a simple explicit formula for turnover reduction when a large number of alphas are traded on the same execution platform and trades are crossed internally. We model turnover reduction via alpha correlations. Then, for a large number of alphas, turnover reduction is related to the largest eigenvalue and the corresponding eigenvector of the alpha correlation matrix.

Abstruct-Pisarenko’s harmonic retrieval (PHR) method is perhaps the first eigenstructure based spectral estimation technique. The basic step in this method is the computation of eigenvector corresponding to the minimum eigenvalue of the autocorrelation matrix of the underlying data. In this paper, we recast a known constrained minimization formulation for obtaining this eigenvector into the neu...