A generalized matrix product can be formally written as A sp p A sp−1 p−1 · · ·A s2 2 A s1 1 , where si ∈ {−1,+1} and (A1, . . . , Ap) is a tuple of (possibly rectangular) matrices of suitable dimensions. The periodic eigenvalue problem related to such a product represents a nontrivial extension of generalized eigenvalue and singular value problems. While the classification of generalized matri...

A powerful method for solving planar eigenvalue problems is the Method of Particular Solutions (MPS), which is also well known under the name “point matching method”. The implementation of this method usually depends on the solution of one of three types of linear algebra problems: singular value decomposition, generalized eigenvalue decomposition, or generalized singular value decomposition. W...

This paper proposes the definition of the generalized Schur complement on nonstrictly diagonally dominant matrices and general H−matrices by using a particular generalized inverse, and then, establishes some significant results on heredity, nonsingularity and the eigenvalue distribution for these generalized Schur complements.

We calculate the Clarke and Michel-Penot subdifferentials of the function which maps a symmetric matrix to its mth largest eigenvalue. We show these two subdifferentials coincide, and are identical for all choices of index m corresponding to equal eigenvalues. Our approach is via the generalized directional derivatives of the eigenvalue function, thereby completing earlier studies on the classi...

Let G be a connected graph with least eigenvalue −2, of multiplicity k. A star complement for −2 in G is an induced subgraph H = G − X such that |X | = k and −2 is not an eigenvalue of H . In the case that G is a generalized line graph, a characterization of such subgraphs is used to decribe the eigenspace of −2. In some instances, G itself can be characterized by a star complement. If G is not...

This paper proposes the definition of the generalized Schur complement on nonstrictly diagonally dominant matrices and general H−matrices by using a particular generalized inverse, and then, establishes some significant results on heredity, nonsingularity and the eigenvalue distribution for these generalized Schur complements.

We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE exhibits a vast improvement over the fastest algorithms we are currently aware of; using a serial PC, it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. The minimi...

This paper continues the early studies on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectral (Math. A unifying framework of creating crucial perturbation equations are developed. With the help of a recent result on generalized commutators involving unitary matrices, new and much sharper bounds are obtained. Abstract This paper continues the early studies on ei...

Abstract We extend to matrix-valued stochastic processes, some well-known relations between realvalued diffusions and classical orthogonal polynomials, along with some recent results about Lévy processes and martingale polynomials. In particular, joint semigroup densities of the eigenvalue processes of the generalized matrix-valued Ornstein-Uhlenbeck and squared OrnsteinUhlenbeck processes are ...

We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE exhibits a vast improvement over the fastest algorithms we are currently aware of; it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. The minimization problem is e...