# Registered Data

## [00505] Structured matrices with applications in sciences and engineering

**Session Date & Time**:- 00505 (1/3) : 5B (Aug.25, 10:40-12:20)
- 00505 (2/3) : 5C (Aug.25, 13:20-15:00)
- 00505 (3/3) : 5D (Aug.25, 15:30-17:10)

**Type**: Proposal of Minisymposium**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.**Organizer(s)**: Susana Furtado, Natália Bebiano**Classification**:__15A09__,__15A29__,__15B48__,__47J20__,__90B50__**Speakers Info**:- Natália Bebiano (Department of Mathematics, University of Coimbra, Portugal)
- Shigeru Furuichi (Department of Information Science, Nihon University, Setagaya-ku, Tokyo, Japan)
**Susana Borges Furtado**(Faculdade de Economia do Porto and CEAFEL)- Wei-Ru Xu (School of Mathematical Sciences, Sichuan Normal University, P.R. China)
- Sirani K. Mututhanthrige Perera (Department of Mathematics, Embry-Riddle Aeronautical University, 1 Aerospace Blvd., Daytona Beach, FL 32114, 386-226-7257)
- Maria isabel Bueno Cachadina (University of California, Santa Barbara)
- Charles R. Johnson (College of William and Mary, Virginia, USA)
- Joao António Ribeiro Cardoso (Polytechnic Institute of Coimbra – ISEC)
- Kenji Toyonaga (Toyohashi University of Technology)
- Koratti Chengalrayan Sivakumar (Indian Institute of Technology Madras)

**Talks in Minisymposium**:**[00811] Singular matrices whose Moore-Penrose inverse is tridiagonal.****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.

**[01723] A matrix approach to the study of efficient vectors in priority setting methodology****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.

**[01819] Computational Techniques for the Mittag-Leffler Function of a Matrix Argument****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.

**[02170] A reduction algorithm for reconstructing periodic pseudo-Jacobi matrices****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.

**[02264] Reciprocal Matrices, Ranking and the Relationship with Social Choice****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.

**[03060] Spectral geometric mean versus geometric mean by generalized Kantorovich constant****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.