This paper gives max characterizations for the sum of the largest eigen-values of a symmetric matrix. The elements which achieve the maximum provide a concise characterization of the generalized gradient of the eigenvalue sum in terms of a dual matrix. The dual matrix provides the information required to either verify rst-order optimality conditions at a point or to generate a descent direction...

This work deals with the generalized eigenvalue problem for nonsquare matrix pencils A − λB such that matrices A,B ∈ MC (m×n) show a given structure. More precisely, we assume they result from removing the first row of some matrix G ∈ MC ((m+1) , n) in the case of A, and its last row in the case of B. This structured generalized eigenvalue problem can be found in signal processing methods and i...

This paper examines the class of generalized Morse wavelets, which are eigenfunction wavelets suitable for use in time-varying spectrum estimation via averaging of time-scale eigenscalograms. Generalized Morse wavelets of order (the corresponding eigenvalue order) depend on a doublet of parameters ( , ); we extend results derived for the special case = = 1 and include a proof of “the resolution...

In this paper an analogue of the Schwarz alternating method is considered to nd a minimal eigenvalue and its corresponding eigenvector of generalized symmetric eigenvalue problem. The technique suggested is based on decomposition of the original domain into overlapping subdomains and on consideration of local eigenvalue problems in subdomains. Both multiplicative and additive variants of the me...

This Thesis contains contributions in two different but closely related subfields of Scientific and Parallel Computing which arise in the context of various eigenvalue problems: periodic and parallel eigenvalue reordering and parallel algorithms for Sylvestertype matrix equations with applications in condition estimation. Many real world phenomena behave periodically, e.g., helicopter rotors, r...

In this report, we reexamine the foundations of classical control theory with reference to eigenvalue geometry via a new set of graphical controls tools, gain plots. These plots depict eigenvalue trajectory as an explicit function of gain. Various classical control concepts are generalized in nine theorems and demonstrated in three simple, yet pedagogical, examples. University Libraries Carnegi...

In this paper, we propose a multi-level finite element method for the transmission eigenvalue problem of anisotropic media. The problem is non-standard and non-self-adjoint with important applications in inverse scattering theory. We employ a suitable finite element method to discretize the problem. The resulting generalized matrix eigenvalue problem is large, sparse and non-Hermitian. To compu...

In a recent paper by Hirzallah et al. [O. Hirzallah, F. Kittaneh, M. Krnić, N. Lovričević, and J. Pečarić. Eigenvalue inequalities for differences of means of Hilbert space operators. Linear Algebra and its Applications, 436:1516–1527, 2012.], several eigenvalue inequalities are obtained for the difference of weighted arithmetic and weighted geometric means of two positive invertible operators ...

Let G be a finite graph of order n with an eigenvalue μ of multiplicity k. (Thus the μ-eigenspace of a (0, 1)-adjacency matrix of G has dimension k.) A star complement for μ in G is an induced subgraph G−X of G such that |X| = k and G−X does not have μ as an eigenvalue. An exceptional graph is a connected graph, other than a generalized line graph, whose eigenvalues lie in [−2,∞). We establish ...

A fast method for enclosing all eigenpairs in symmetric positive definite generalized eigenvalue problems is proposed. Firstly theorems on verifying all eigenvalues are presented. Next a theorem on verifying all eigenvectors is presented. The proposed method is developed based on these theorems. Numerical results are presented showing the efficiency of the proposed method. As an application of ...