Abstract : The main purpose of this MS is to present recent developments on some special structured matrices that are of interest in different areas of mathematics, as well as in more applied areas like operations research, social sciences and computation. Problems arising in these fields are considered and techniques from matrix theory, numerical linear algebra and combinatorics, among others, are explored to solve them.
[02264] Reciprocal Matrices, Ranking and the Relationship with Social Choice
Format : Online Talk on Zoom
Author(s) :
Charles R Johnson (William and Mary)
Abstract : There is a close connection between the use of efficient vectors for reciprocal (pairwise comparison) matrices, used
in business project ranking schemes, and social choice/voting rules from political science and economics. However,
the two seem not to have been discussed together before. We explore this connection, as well as advance the theory
of reciprocal matrices. In addition, there seem to be natural connections with other parts of economic theory.
[01723] A matrix approach to the study of efficient vectors in priority setting methodology
Format : Talk at Waseda University
Author(s) :
Susana Furtado (Faculdade de Economia do Porto and CEAFEL)
Charles Johnson (College of William and Mary)
Abstract : The Analytic Hierarchy Process is a much discussed method in ranking business alternatives based on empirical and judgemental information.
Here we use a matrix approach to study the key component of efficient vectors for a reciprocal matrix of pairwise comparisons. In particular, we give new efficient vectors for a reciprocal matrix, which we compare numerically with other known efficient vectors.
[00811] Singular matrices whose Moore-Penrose inverse is tridiagonal.
Format : Talk at Waseda University
Author(s) :
Maria Isabel Bueno Cachadina (University of California Santa Barbara)
Susana Borges Furtado (Faculdade de Economia do Porto and CEAFEL)
Abstract : A variety of characterizations of nonsingular matrices whose inverse is
tridiagonal (irreducible or not) have been widely investigated in the
literature. One well-known such characterization is stated in terms of
semiseparable matrices. In this talk, we consider singular matrices $A$ and
give necessary and sufficient conditions for the Moore-Penrose inverse of $A$
to be tridiagonal. Our approach is based on bordering techniques, as given by
Bapat and Zheng (2003). In addition, we obtain necessary conditions on $A$
analogous to the semiseparability conditions in the nonsingular case, though
in the singular case they are not sufficient, as illustrated with examples. We
apply our results to give an explicit description of all the $3\times3$ real
singular matrices and $3\times3$ Hermitian matrices whose Moore-Penrose
inverse is irreducible and tridiagonal.
[03060] Spectral geometric mean versus geometric mean by generalized Kantorovich constant
Format : Talk at Waseda University
Author(s) :
Shigeru Furuichi (Nihon University)
Abstract : In this talk, we give two different operator inequalities between the weighted spectral geometric mean and the weighted geometric mean. We also study the mathematical properties for the generalized Kantorovich constant. Applying the obtained inequalities on the generalized Kantorovich constant, we give the ordering of two inequalities between the weighted spectral geometric mean and the weighted geometric mean.
In addition, we give some inequalities such as Ando type inequality, Kantorovich type inequality, and Ando-Hiai type inequality with the weighted spectral geometric mean and the generalized Kantorovich constant.
[05539] Bundles of matrix pencils under strict equivalence
Format : Talk at Waseda University
Author(s) :
FERNANDO DE TERÁN (Universidad Carlos III de Madrid)
Froilán Martínez Dopico (Universidad Carlos III de Madrid)
Abstract : Bundles of matrix pencils are sets of pencils having the same Kronecker canonical form, up to the eigenvalues (namely, they are a union of orbits under strict equivalence). This notion was introduced in the 1990’s, following the one for matrices under similarity (from Arnold, 1971). In this talk, we provide a characterization for the inclusion relation between closures of bundles and prove that bundles are open in their closure (in the standard topology) .
[05542] Row completion of polynomial matrices
Format : Talk at Waseda University
Author(s) :
Alicia Roca (Universitat Politècnica de València / IMM, Valencia, Spain)
Agurtzane Amparan (Universidad del País Vasco UPV/EHU)
Itziar Baragaña (Universidad del País Vasco UPV/EHU)
Silvia Marcaida (University of the Basque CountryUniversidad del País Vasco UPV/EHU)
Abstract : Perturbation problems arise frequently in applications, as in structural changes of the dynamics of a system or in pole placement problems in control theory.
Perturbation problems of matrices are closely related to completion problems. We present a solution to the row-completion problem of a polynomial matrix, prescribing the eigenstructure of the resulting matrix and maintaining the degree.
[01819] Computational Techniques for the Mittag-Leffler Function of a Matrix Argument
Format : Talk at Waseda University
Author(s) :
João R. Cardoso (Polytechnic Institute of Coimbra – ISEC)
Abstract : It is well-known that the two-parameter Mittag-Leffler function plays a key role in Fractional Calculus. In this talk, we address the problem of computing this function, when its argument is a square matrix. Effective methods for solving this problem involve the computation of successive derivatives or require the use of mixed precision arithmetic. We provide an alternative method that is derivative-free and can work entirely using IEEE standard double precision arithmetic. Our method starts with a reordered Schur decomposition of the argument matrix, so that the problem reduces to the computation of the Mittag-Leffler function of a triangular matrix with ``close'' eigenvalues. Theoretical and numerical issues regarding the performance of the method are investigated. A set of numerical experiments show that our novel approach is competitive with the existing ones, in terms of accuracy and computational cost.
[05653] Classification of edges due to the change in multiplicity of an eigenvalue
Format : Talk at Waseda University
Author(s) :
KENJI TOYONAGA (Toyohashi University of Technology)
Abstract : We give possible classifications of edges in a general undirected graph in terms of the change in multiplicity of an eigenvalue by removing the edge.
Further, we give a characterization of Parter vertices associated with the downer branch mechanism in general graphs. When the graph is a tree, the existence of a downer branch at a Parter vertex has been known in the previous work. We clarify the downer branch mechanism, Then we give the effect for the classifications of other edges or vertices in the remaining graph by removing a 2-downer edge.
[02170] A reduction algorithm for reconstructing periodic pseudo-Jacobi matrices
Format : Online Talk on Zoom
Author(s) :
Natalia Bebiano (Department of Mathematics, Coimbra University)
Abstract : For the given signature operator $\mathcal{H}=I_{r}\oplus-I_{n-r}$, a pseudo-Jacobi matrix is a self-adjoint matrix relatively to a symmetric bilinear form $\langle \cdot,\cdot\rangle_{\mathcal{H}}$. In this talk, we consider recent inverse eigenvalue problems for this class of matrices. Necessary and sufficient conditions under which the problems have solution are presented. Numerical algorithms are designed according to the obtained theoretical results. Illustrative numerical examples are given to test the reconstructive algorithms.
[05469] A Low-Cost Algorithm to Determine Orbital Trajectories
Format : Talk at Waseda University
Author(s) :
Sirani M. Perera (Embry-Riddle Aeronautical University)
David Canales (Embry-Riddle Aeronautical University)
Brian Baker-McEvilly (Embry-Riddle Aeronautical University)
Abstract : The increasing demand for effective methods to propagate trajectories in the circular restricted three-body problem (CR3BP) is driven by heavy traffic in the region. This presentation introduces a
novel approach that utilizes interpolation for determining orbital trajectories. We present a novel algorithm for determining orbital motion in the CR3BP, which is specifically applied to different
Cislunar trajectories. Our findings demonstrate a 50% reduction in time complexity compared to the existing methods.
