Abstract : The goal of the minisymposium is to stimulate research and foster interaction of researchers. It scope includes any topics in matrices and their applications. Matrix analysis is widely used in mathematics with applications in control and systems theory, image processing, operations research, scientific computing, statistics, and engineering. This minisymposium has multiple sessions which provide an opportunity for researchers to exchange ideas and recent developments in this active area of research. Participants are from Canada, China, Japan, Macau, Norway, Pakistan, Portugal, Singapore, South Korea, and USA.
Organizer(s) : Luyining Gan, Tin-Yau Tam, Qing-Wen Wang, Yang Zhang
Xin Liu (Macau University of Science and Technology)
Rute Lemos (University of Aveiro)
Mutti-Ur Rehman (Sukkur IBA University)
Takeaki Yamazaki (Toyo University)
Huihui Zhu (Hefei University of Technology)
Tin-Yau Tam (University of Nevada, Reno)
Yang Zhang (University of Manitoba)
Qing-Wen Wang (Shanghai University)
Talks in Minisymposium :
[00800] Limit of the induced Aluthge transformations
Author(s) :
Takeaki Yamazaki (Toyo University)
Abstract : Let $\mathcal{H}$ and $B(\mathcal{H})$ be a complex Hilbert space and the algebra of all bounded linear operators on $\mathcal{H}$, respectively. For $T\in B(\mathcal{H})$ with the polar decomposition $T=U|T|$, Aluthge transformation is known as $\Delta(T):=|T|^{1/2}U|T|^{1/2}$. In this talk, we shall introduce a generalization of Aluthge transformations and limit point of its iterations.
[00804] (B,C)-MP inverse of a rectangular matrix
Author(s) :
Huihui Zhu (Hefei University of Technology)
Abstract : Let $\mathbb{C}^{m\times n}$ be the set of $m\times n $ complex matrices and let $A\in \mathbb{C}^{n\times m}$ and $B,C\in \mathbb{C}^{m\times n}$ such that the $(B,C)$-inverse of $A$ (denoted by $A^{\|(B,C)}$) exists. A matrix $A$ is called $(B,C)$-Moore-Penrose (abbr. $(B,C)$-MP-inverse) invertible if there exists some $X\in \mathbb{C}^{m\times n}$ such that $XAX=X, XA=A^{\|(B,C)}A$ and $CAX=CAA^\dag$. Such an $X$ is called a $(B,C)$-MP-inverse of $A$. Several characterizations for the $(B,C)$-MP-inverse of $A$ are obtained. Also, the connection between $(B,C)$-MP-inverses and other generalized inverses is also given. Finally, we determine the rank of $AA^{\|(B_{1},C_{1}),\dag}-DD^{\|(B_{2},C_{2}),\dag}$.
[00807] Images of polynomial derivations and LFED conjecture
Author(s) :
Xiankun Du (Jilin University)
Hongyu Jia (Jilin University)
Haifeng Tian (Jilin University)
Abstract : We prove that images of linear derivations and linearE-derivations of $K[x,y,z]$ and E-derivations of $K[x,y]$ are Mathieu-Zhao subspaces.
[00830] The ranks and decompositions of quaternion tensors
Author(s) :
Yang Zhang (University of Manitoba)
Yungang Liang (University of Manitoba)
Abstract : Quaternion tensors have attracted more and more attentions in recent years. Many applications have been found in various areas. In this talk, we discuss the maximal ranks of quaternion tensors, in particular, the third-order case. We also investigate the canonical forms, CP and Tucker decompositions of some quaternion tensors.
[00868] Matrix Problems in International Economics
Author(s) :
Konstantin Kucheryavyy (University of Tokyo)
Abstract : Analyzing properties of general equilibrium models in economics amounts to analyzing properties of nonlinear systems of equations. The standard questions asked in this context are about existence and uniqueness of solutions to a nonlinear system of equations, characterization of multiplicity of solutions, and behavior of solutions in response to changes in parameters. One approach to address these questions is to formulate them in terms of matrix algebra. This approach has been especially fruitful when applied to classes of models arising in international economics. Such models often give rise to matrices with striking properties, but formally proving the observed properties usually constitutes a challenge. In my presentation, I will consider several matrix problems that arise in the international economics context and sketch related proofs. I will also discuss open questions in this literature.
[01222] Some new results on matrix and tensor equations
Author(s) :
Qing-Wen Wang (Shanghai University, China)
Abstract : In this talk, we mainly introduce some new developments of matrix and tensor equations over the quaternion algebra.
[01238] A Descent Method for Symmetric Nonnegative Matrix Factorization
Author(s) :
Delin Chu (National University of Singapore, SingaporeNational University of Singapore)
Liangshao Hou (Hong Kong Baptist University)
Li-Zhi Liao (Hong Kong Baptist University)
Abstract : In this talk, the symmetric nonnegative matrix factorization (SNMF), which is a powerful tool in data mining for data dimension reduction and clustering, is discussed. Our present work is introduced including: (i) a new descent direction for the rank-one SNMF is derived and a strategy for choosing the step size along this descent direction is established; (ii) a progressive hierarchical alternating least squares (PHALS) method for SNMF is developed, which is parameter-free and updates the variables column by column. Moreover, every column is updated by solving a rank-one SNMF subproblem; and (iii) the convergence to the Karush-Kuhn-Tucker (KKT) point set (or the stationary point set) is proved for PHALS. Several synthetical and real data sets are tested to demonstrate the effectiveness and efficiency of the proposed method. Our PHALS provides better performance in terms of the computational accuracy, the optimality gap, and the CPU time, compared with a number of state-of-the-art SNMF methods.
[01263] Geometric inequalities for contraction matrices
Author(s) :
Tin-Yau Tam (University of Nevada, Reno)
Abstract : Given two $n\times n$ contraction matrices $W$ and $Z$, i.e., $I-WW^*\ge 0$ and $I-ZZ^*\ge 0$, L.K. Hua's inequalities (1955) assert that
$$
\det (I-WW^*)\det (I-ZZ^*) \le |\det (I - WZ^*)|^2\le \det (I+WW^*)\det (I+ZZ^*).
$$
In this talk we will present geometry behind Hua's inequalities in the context of elliptical and hyperbolic geometry.
[01266] Majorization orders for $(0,\pm 1)$-matrices
Author(s) :
Geir Dahl (University of Oslo)
Alexander Guterman (Bar-Ilan University)
Pavel Shteyner (Bar-Ilan University)
Abstract : Matrix majorization is a generalization of classical majorization for vectors; an important notion in many areas of mathematics. The talk gives some majorization background, and then presents a study of matrix majorization for $(0, \pm 1)$-matrices, i.e., matrices whose entries are restricted to $0$, $1$ and $-1$. In particular, we characterize when the zero vector is weakly majorized by a matrix, and discuss related results. Different connections are discussed, and characterizations of majorization are given.
[01267] Combinatorial Perron Parameters and Classes of Trees
Author(s) :
Enide Cascais Andrade (CIDMA, University of Aveiro, Portugal)
Lorenzo Ciardo (University of Oxford)
Geir Dahl (University of Oslo)
Abstract :
The main goal of this talk is to present recent results related with the combinatorial Perron
parameters
introduced in previous papers for certain classes of trees, and related bounds for these parameters.
These parameters are related to algebraic connectivity of trees and corresponding centers.
[01307] The generalized quaternion matrix equation
Author(s) :
Xin Liu (Macau University of Science and Technology)
Cui E Yu (Macau University of Science and Technology)
Abstract : We consider the matrix equation $AXB+CX^{\star}D=E$ over the generalized quaternions, where $X^\star$ is one of $X$, $X^\ast$, the $\eta$-conjugate or the $\eta$-conjugate transpose of $X$ with $\eta \in \{i, j, k\}$. We define two new real representations of a generalized quaternion matrix, then we derive the solvability conditions for the mentioned matrix equation. Moreover, we also discuss the existence of $X=\pm X^{\star}$ solutions to the generalized quaternion matrix equation $AXB+CXD=E$.
[01363] On the Hurwitz stability and quasi-stability of matrix polynomials
Author(s) :
Yongjian Hu (Beijing Normal University)
Xuzhou Zhan (Beijing Normal University at Zhuhai)
Abstract : This talk is concerned with the Hurwitz stability and quasi-stability of matrix polynomials. Two methods are adopted to deduce the stability tests: one is an algebraic method based on the use of the linear structured features of the Markov parameters, while the other is associated with the analytic properties of the rational matrix-valued functions. Some classical theorems by Gantmacher, Hermite, Biehler, et al. are generalized to the matricial case as well.
[01379] Poset matrices and associated algebras
Author(s) :
Gi-Sang Cheon (Sungkyunkwan University)
Abstract : We introduce the constructions of poset matrices by defining several partial compositions on the species of poset matrices. Some of these partial composition operations are shown to define a set operad structure. We also obtain various matrix algebras obtained from incidence algebras of Riordan posets.
[01404] Spectral inequalities for Kubo-Ando and Heinz means
Author(s) :
Rute Correia Lemos (CIDMA, University of Aveiro)
Graça Soares (CMAT-UTAD, University of Trás-os Montes e Alto Douro)
Abstract : In this talk, spectral inequalities, involving Kubo-Ando operator connections and means of positive semidefinite matrices, are surveyed. Some Log-majorization type results are presented. Singular values inequalities for Heinz mean of matrices, which are not of Kubo-Ando type, and its 'harmonic' variant are also given.
[01920] Matrix inequalities and properties of means on positive definite matrices
Author(s) :
Luyining Gan (University of Nevada Reno)
Abstract : In this talk, we will introduce the study of the relations between the weighted metric geometric mean, the weighted spectral geometric mean and the weighted Wasserstein mean of the positive definite matrices in terms of (weak) log-majorization relation. In addition, we will also introduce some new properties of means, like geodesic property and tolerance relation.
[03621] Log-majorization and inequalities of power means
Author(s) :
Sejong Kim (Chungbuk National University)
Abstract : As non-commutative versions of the quasi-arithmetic mean, we consider the Lim-Palfia’s power mean, Renyi right mean, and Renyi power mean of positive definite matrices. We see that the Lim-Palfia’s power mean of negative order converges increasingly to the log-Euclidean mean with respect to the weak log-majorization. Furthermore, we establish the weak log-majorization relationships between power means and provide the boundedness of Renyi power mean.
