We address the homogenization of an eigenvalue problem for the neutron transport equation in a periodic heterogeneous domain, modeling the criticality study of nuclear reactor cores. We prove that the neutron flux, corresponding to the first and unique positive eigenvector, can be factorized in the product of two terms, up to a remainder which goes strongly to zero with the period. One term is ...

The least-squares method can be used to approximate an eigenvector for a matrix when only an approximation is known for the corresponding eigenvalue. In this paper, this technique is analyzed and error estimates are established proving that if the error in the eigenvalue is sufficiently small, then the error in the approximate eigenvector produced by the least-squares method is also small. Also...

In many engineering applications, the physical quantities that have to be computed are obtained by solving a related eigenvalue problem. The matrix under consideration and thus its eigenvalues usually depend on some parameters. A natural question then is how sensitive the physical quantity is with respect to (some of) these parameters, i.e., how it behaves for small changes in the parameters. T...

The spectrum of a square matrix A, denoted by σ(A) is the set of all eigenvalues of A. The spectral radius of A, denoted by ρ(A) is defined as: ρ(A) = max{|λ| : λ ∈ σ(A)} An eigenvalue of A that is larger in absolute value than any other eigenvalue is called the dominant eigenvalue; a corresponding eigenvector is called a dominant eigenvector. The most commonly encountered vector norm (often si...

This paper presents the applications of Eigenvalues and Eigenvectors (as part of spectral decomposition) to analyze the bipartivity index of graphs as well as to predict the set of vertices that will constitute the two partitions of graphs that are truly bipartite and those that are close to being bipartite. Though the largest eigenvalue and the corresponding eigenvector (called the principal e...

We consider polynomial eigenvalue problems P(A; ;)x = 0 in which the matrix polynomial is homogeneous in the eigenvalue (;) 2 C 2. In this framework innnite eigenvalues are on the same footing as nite eigenvalues. We view the problem in projective spaces to avoid normalization of the eigenpairs. We show that a polynomial eigenvalue problem is well-posed when its eigenvalues are simple. We deene...

We consider polynomial eigenvalue problems P(A, α, β)x = 0 in which the matrix polynomial is homogeneous in the eigenvalue (α, β) ∈ C2. In this framework infinite eigenvalues are on the same footing as finite eigenvalues. We view the problem in projective spaces to avoid normalization of the eigenpairs. We show that a polynomial eigenvalue problem is wellposed when its eigenvalues are simple. W...

This note gives converses to the well-known result: for any vector e u such that sin (u; e u) = O( ), we have e u Ae u e u e u = + O( ) where is an eigenvalue and u is the corresponding eigenvector of a Hermitian matrix A, and \ " denotes complex conjugate transpose. It shows that if e u Ae u=e u e u is close to A's largest eigenvalue, then e u is close to the corresponding eigenvector with an ...

While several methods aimed at understanding the causes of model behavior have been proposed in recent years, formal model analysis remains an important and challenging area in system dynamics. This paper describes a mathematical method to incorporate eigenvectors to the more traditional eigenvalue analysis of dynamic models. The proposed method derives basic formulas that characterize how a ch...