Abstract The eigenvector-eigenvalue identities areexpanded to include general mixing parameters. Some simple relations are obtained and they reveal an intricate texture of connections between the eigenvalues Permutation symmetry ( $$S_{3}\times S_{3}$$ S 3 ×</mml...

Given an n-by-n Hermitian matrix A and a real number λ, index i is said to be Parter (resp. neutral, downer) if the multiplicity of λ as an eigenvalue of A(i) is one more (resp. the same, one less) than that in A. In case the multiplicity of λ in A is at least 2 and the graph of A is a tree, there are always Parter vertices. Our purpose here is to advance the classification of vertices and, in ...

This paper concerns with the sensitivity analysis for the multivariate eigenvalue problem (MEP). The concept of a simple multivariate eigenvalue of a matrix is generalized to the MEP and the first-order perturbation expansions of a simple multivariate eigenvalue and the corresponding multivariate eigenvector are presented. The explicit expressions of condition numbers, perturbation upper bounds...

A complete characterization of the convergence factor can be very useful when analyzing the asymptotic convergence of an iterative method. We will here establish a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well known inverse iteration. The formula for the convergence factor ...

According to metapopulation theory, the capacity of a habitat patch network to support the persistence of a species is measured by the metapopulation capacity of the patch network. Mathematically, metapopulation capacity is given by the leading eigenvalue kM of an appropriately constructed non-negative n n matrix M, where n is the number of habitat patches. Both habitat destruction (in the sens...

Let G be a connected simple graph. The relationship between the third smallest eigenvalue of the Laplacian matrix and the graph structure is explored. For a tree the complete description of the eigenvector corresponding to this eigenvalue is given and some results about the multiplicity of this eigenvalue are given.

We present an efficient method for preparing the initial state required by the eigenvalue approximation quantum algorithm of Abrams and Lloyd. Our method can be applied when solving continuous Hermitian eigenproblems, e.g., the Schrödinger equation, on a discrete grid. We start with a classically obtained eigenvector for a problem discretized on a coarse grid, and we efficiently construct, quan...

This paper derives a posteriori error estimates for conforming numerical approximations of the Laplace eigenvalue problem with a homogeneous Dirichlet boundary condition. In particular, upper and lower bounds for an arbitrary simple eigenvalue are given. These bounds are guaranteed, fully computable, and converge with optimal speed to the given exact eigenvalue. They are valid without restricti...

This paper concerns with the sensitivity analysis for the multivariate eigenvalue problem (MEP). The concept of a simple multivariate eigenvalue of a matrix is generalized to the MEP and the first-order perturbation expansions of a simple multivariate eigenvalue and the corresponding multivariate eigenvector are presented. The explicit expressions of condition numbers, perturbation upper bounds...