Positive Eigenvector Research Articles

The Principal Eigenvalue for Degenerate Periodic Reaction-Diffusion Systems

A generalized Krein-Rutman theorem for a strongly positive bounded linear operator whose spectral radius is larger than essential spectral radius is established: the spectral radius of the operator is an algebraically simple eigenvalue with strongly positive eigenvector and other eigenvalues are less than the spectral radius.

Positive eigenvector of nonlinear eigenvalue problem with a singular M-matrix and Newton-SOR iterative solution

Some sufficient conditions are proposed in this paper such that the nonlinear eigenvalue problem with an irreducible singular M-matrix has a unique positive eigenvector. Under these conditions, the Newton-SOR iterative method is proposed for numerically solving such a positive eigenvector and some convergence results on this iterative method are established for the nonlinear eigenvalue problems with an irreducible singular M-matrix, a nonsingular M-matrix, and a general M-matrix, respectively. Finally, a numerical example is given to illustrate that the Newton-SOR iterative method is superior to the Newton iterative method.

On the principal eigenvectors of uniform hypergraphs

Let A(H) be the adjacency tensor of r-uniform hypergraph H. If H is connected, the unique positive eigenvector x=(x1,x2,⋯,xn)T with ‖x‖r=1 corresponding to spectral radius ρ(H) is called the principal eigenvector of H. The maximum and minimum entries of x are denoted by xmax and xmin, respectively. In this paper, we investigate the bounds of xmax and xmin in the principal eigenvector of H. Meanwhile, we also obtain some bounds of the ratio xi/xj for i, j∈[n] as well as the principal ratio γ(H)=xmax/xmin of H. As an application of these results we finally give an estimate of the gap of spectral radii between H and its proper sub-hypergraph H′.

Weapon systems accuracy evaluation using the error spectrum

Weapon systems accuracy is a vitally important tactical and technical index, which has a significant impact on the process of the weapon systems design and test. Based on different accuracy criteria, we proposed a new evaluation method for the weapon systems accuracy. First, a new bootstrap method, using the correlation coefficient criterion, is presented to evaluate a weapon systems accuracy when the sample size n satisfied n≥10. Second, measures for weapon systems accuracy are introduced to represent the different performance of weapon systems. Furthermore, an attributes matrix is proposed to combine the above measures. Third, inspired by the Pitman's closeness measure, we designed a weapon systems accuracy attributes competition matrix to contain the entire pairwise competition results, and then its positive eigenvector is employed to the weapon systems accuracy evaluation. Finally, a numerical example is provided to illustrate the effectiveness and feasibility of the proposed method. It is shown that the new evaluation method can not only evaluate the weapon systems accuracy before and after improvement, but also rank the accuracy of the designed weapon systems to the evaluated weapon systems.

A note on the eigenvectors of perturbed matrices with applications to linear positive systems

A result is presented describing the eigenvectors of a perturbed matrix, for a class of structured perturbations. One motivation for doing so is that positive eigenvectors of nonnegative, irreducible matrices are known to induce norms — acting much like Lyapunov functions — for linear positive systems, which may help estimate or control transient dynamics. The results apply to both discrete- and continuous-time linear positive systems. The theory is illustrated with several examples.

The local spectral radius of a nonnegative orbit of compact linear operators

Abstract We consider orbits of compact linear operators in a real Banach space which are nonnegative with respect to the partial ordering induced by a given cone. The main result shows that under a mild additional assumption the local spectral radius of a nonnegative orbit is an eigenvalue of the operator with a positive eigenvector.

Continuity properties of the semi-group and its integral kernel in non-relativistic QED

Employing recent results on stochastic differential equations associated with the standard model of non-relativistic quantum electrodynamics by B. Güneysu, J. S. Møller, and the present author, we study the continuity of the corresponding semi-group between weighted vector-valued [Formula: see text]-spaces, continuity properties of elements in the range of the semi-group, and the pointwise continuity of an operator-valued semi-group kernel. We further discuss the continuous dependence of the semi-group and its integral kernel on model parameters. All these results are obtained for Kato decomposable electrostatic potentials and the actual assumptions on the model are general enough to cover the Nelson model as well. As a corollary, we obtain some new pointwise exponential decay and continuity results on elements of low-energetic spectral subspaces of atoms or molecules that also take spin into account. In a simpler situation where spin is neglected, we explain how to verify the joint continuity of positive ground state eigenvectors with respect to spatial coordinates and model parameters. There are no smallness assumptions imposed on any model parameter.

A non-linear non-weight method for multi-criteria decision making

We apply the Perron theorem in multi-attribute decision making. We create a comparison matrix for decision alternatives and prove that the matrix is almost-always primitive. We use the limiting power of the matrix multiplied by a standard vector, which leads to a positive eigenvector of the matrix, as the ranking vector for decision alternatives. The proposed method does not require domain experts to assign weights for decision criteria as usually demanded by the weighted-sum model. The new method is simple to use and generates reasonable result as illustrated by an example of ranking best hospitals over twelve criteria. We also demonstrate that the weightedsum methods may not be able to reveal all possible rankings. We give one example showing that a weighted-sum method collapsed thirteen distinct rankings into a single ranking and another example showing that the weighted-sum methods could not produce the ranking that is unrenderable by linear functions.

$${{C^m}}$$ C m Positive Eigenvectors for Linear Operators Arising in the Computation of Hausdorff Dimension

We consider a broad class of linear Perron–Frobenius operators $${\Lambda:X \rightarrow X}$$ , where $${X}$$ is a real Banach space of $${C^m}$$ functions. We prove the existence of a strictly positive $${C^m}$$ eigenvector $${v}$$ with eigenvalue $${r=r(\Lambda) =}$$ the spectral radius of $${\Lambda}$$ . We prove (see Theorem 6.5 in Sect. 6 of this paper) that $${r(\Lambda)}$$ is an algebraically simple eigenvalue and that, if $${\sigma(\Lambda)}$$ denotes the spectrum of the complexification of $${\Lambda,\sigma(\Lambda) \backslash \{r(\Lambda)\} \subseteq \{\zeta \in \mathbb{C} \big| |\zeta| \le r_*\}}$$ , where $${r_* < r(\Lambda)}$$ . Furthermore, if $${u \in X}$$ is any strictly positive function, $${(\frac 1r \Lambda)^k(u) \rightarrow s_u v}$$ as $${k \rightarrow \infty}$$ , where $${s_u > 0}$$ and convergence is in the norm topology on $${X}$$ . In applications to the computation of Hausdorff dimension, one is given a parametrized family $${\Lambda_s,s > s_*}$$ , of such operators and one wants to determine the (unique) value $${s_0}$$ such that $${r(\Lambda_{s_0})=1}$$ . In another paper (Falk and Nussbaum in C $${^{\rm m}}$$ Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension, submitted) we prove that explicit estimates on the partial derivatives of the positive eigenvector $${v_s}$$ of $${\Lambda_s}$$ can be obtained and that this information can be used to give rigorous, sharp upper and lower bounds for $${s_0}$$ .

Taking All Positive Eigenvectors Is Suboptimal in Classical Multidimensional Scaling

It is hard to overstate the importance of multidimensional scaling as an analysis technique in the broad sciences. Classical, or Torgerson multidimensional scaling is one of the main variants, with the advantage that it has a closed-form analytic solution. However, this solution is exact if and only if the distances are Euclidean. Conversely, there has been comparatively little discussion on what to do in the presence of negative eigenvalues: the intuitive solution, prima facie justifiable in least-squares terms, is to take every positive eigenvector as a dimension. We show that this, minimizing least-squares to the centred distances instead of the true distances, is suboptimal - throwing away positive eigenvectors can decrease the error even as we project to fewer dimensions. We provide provably better methods for handling this common case.

A Necessary and Sufficient Condition for Existence of a Positive Perron Vector

In 1907, Perron showed that a positive square matrix has a unique largest positive eigenvalue with a positive eigenvector. This result was extended to irreducible nonnegative matrices by Frobenius in 1912, and to irreducible nonnegative tensors and weakly irreducible nonnegative tensors recently. This result is a fundamental result in matrix theory and has found wide applications in probability theory, internet search engines, spectral graph and hypergraph theory, etc. In this paper, we give a necessary and sufficient condition for the existence of such a positive eigenvector, i.e., a positive Perron vector, for a nonnegative tensor. We show that every nonnegative tensor has a canonical nonnegative partition form, from which we introduce strongly nonnegative tensors. A tensor is called strongly nonnegative if the spectral radius of each genuine weakly irreducible block is equal to the spectral radius of the tensor, which is strictly larger than the spectral radius of any other block. We prove that a nonne...

99.30 The positive eigenvector of a positive matrix

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Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks

Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. Perron-Frobenius theory ensures the existence of a corresponding positive eigenvector $\varphi$. The goal of the paper is to give bounds on the amplitude $\max \varphi/\min\varphi$. Two approaches are proposed: one using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative study of convergence to quasi-stationarity to the convergence to equilibrium of related ergodic processes, as seen in [7].

Index theory-based algorithm for the gradiometer inverse problem

We present an Index Theory based gravity gradiometer inverse problem algorithm. This algorithm relates changes in the index value computed on a closed curve containing a line field generated by the positive eigenvector of the gradiometer tensor to the closeness of fit of the proposed inverse solution to the mass and center of mass of the unknown. We then derive a method of determining bounds on the unknown’s center of mass and/or total mass and apply it as a function of gradiometer observables. Both observational errors and the varieties of possible mass distributions generating the gradients are taken into account for the bounds.

Quasistochastic matrices and Markov renewal theory

Let 𝓈 be a finite or countable set. Given a matrix F = (Fij)i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (qij)i,j∈𝓈, i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑n≥0Qn ⊗ F*n associated with Q ⊗ F := (qijFij)i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(Mn, Sn)}n≥0 with discrete recurrent driving chain {Mn}n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

Quasistochastic matrices and Markov renewal theory

Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

A positivity preserving inexact Noda iteration for computing the smallest eigenpair of a large irreducible $$M$$ M -matrix

In this paper, based on the Noda iteration, we present inexact Noda iterations (INI), to find the smallest eigenvalue and the associated positive eigenvector of a large irreducible nonsingular $$M$$M-matrix. The positivity of approximations is critical in applications, and if the approximations lose the positivity then they may be meaningless and could not be interpreted. We propose two different inner tolerance strategies for solving the inner linear systems involved, and prove that the convergence of resulting INI algorithms is globally linear and superlinear with the convergence order $$\frac{1+\sqrt{5}}{2}$$1+52, respectively. The proposed INI algorithms are structure preserving and maintains the positivity of approximate eigenvectors. We also revisit the exact Noda iteration and establish a new quadratic convergence result. All the above is first done for the problem of computing the Perron root and the positive Perron vector of an irreducible nonnegative matrix and is then adapted to computing the smallest eigenpair of the irreducible nonsingular $$M$$M-matrix. Numerical examples illustrate that the proposed INI algorithms are practical, and they always preserve the positivity of approximate eigenvectors. We compare them with the Jacobi---Davidson method, the implicitly restarted Arnoldi method and the explicitly restarted Krylov---Schur method, all of which cannot guarantee the positivity of approximate eigenvectors, and illustrate that the overall efficiency of the INI algorithms is competitive with and can be considerably higher than the latter three methods.

Nonlinear extensions of the Perron–Frobenius theorem and the Krein–Rutman theorem

A unification version of the Perron–Frobenius theorem and the Krein–Rutman theorem for increasing, positively 1-homogeneous, compact mappings is given on ordered Banach spaces without monotonic norm. A Collatz-type minimax characterization of the positive eigenvalue with positive eigenvector is obtained. The power method in computing the largest eigenpair is also extended.

Positive eigenvalue-eigenvector of nonlinear positive mappings

We show that an (eventually) strongly increasing and positively homogeneous mapping T defined on a Banach space can be turned into an Edelstein contraction with respect to Hilbert’s projective metric. By applying the Edelstein contraction theorem, a nonlinear version of the famous Krein-Rutman theorem is presented, and a simple iteration process {Tkx/‖Tkx‖} (∀ x ∈ P+) is given for finding a positive eigenvector with positive eigenvalue of T. In particular, the eigenvalue problem of a nonnegative tensor A can be viewed as the fixed point problem of the Edelstein contraction with respect to Hilbert’s projective metric. As a result, the nonlinear Perron-Frobenius property of a nonnegative tensor A is reached easily.

On Spectral Hypergraph Theory of the Adjacency Tensor

We study both H and E/Z-eigenvalues of the adjacency tensor of a uniform multi-hypergraph and give conditions for which the largest positive H or Z-eigenvalue corresponds to a strictly positive eigenvector. We also investigate when the E-spectrum of the adjacency tensor is symmetric.