We introduce a cross eigenvalue condition matrix for measuring the sensitivity of eigenvalue computations. A fundamental difference between this new object and traditional eigenvalue condition numbers is that cross interactions between eigenvectors that are not associated with the same eigenvalue are taken into account. We develop an abstract formulation that makes it possible to define continu...

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 ...

The ‘monodromy conjecture’ for a hypersurface singularity f predicts that a pole of its topological (or related) zeta function induces one of its monodromy eigenvalues. However, in general only a few eigenvalues are obtained this way. The second author proposed to consider zeta functions associated with the hypersurface and with a differential form and raised the following question. Can one fin...

the stability analysis of islanded inverter-based microgrids (ibmgs) is increasingly an important and challenging topic due to the nonlinearity of ibmgs. in this paper, a new linear model for such microgrids as well as an iterative method to correct the linear model is proposed. using the linear model makes it easy to analyze the eigenvalues and stability of ibmgs due to the fact that it derive...

In this paper, only simple graphs are considered. A graphG is nonsingular if its adjacency matrix A(G) is nonsingular. A nonsingular graph G satisfies reciprocal eigenvalue property (property R) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property (property SR) if the reciprocal of each eigenvalue o...

Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions and is of practical interest because of a wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the smallest eigenvalue of an analytic and...

In this paper, we examine the maximum eigenvalue function of an even order real symmetric tensor. By using the variational analysis techniques, we first show that the maximum eigenvalue function is a continuous and convex function on the symmetric tensor space. In particular, we obtain the convex subdifferential formula for the maximum eigenvalue function. Next, for an mth-order n-dimensional s...

The stability analysis of islanded inverter-based microgrids (IBMGs) is increasingly an important and challenging topic due to the nonlinearity of IBMGs. In this paper, a new linear model for such microgrids as well as an iterative method to correct the linear model is proposed. Using the linear model makes it easy to analyze the eigenvalues and stability of IBMGs due to the fact that it derive...