Ranks of the common solution to six quaternion matrix equations

2011 ◽  
Vol 27 (3) ◽  
pp. 443-462 ◽  
Author(s):  
Qing-wen Wang ◽  
Yan Zhou ◽  
Qin Zhang
Keyword(s):  
Matrix Equations ◽  
Quaternion Matrix ◽  
Common Solution ◽  
The Common

The common solution to six quaternion matrix equations with applications

2008 ◽  
Vol 198 (1) ◽  
pp. 209-226 ◽  
Author(s):  
Qing-Wen Wang ◽  
Hai-Xia Chang ◽  
Qun Ning
Keyword(s):  
Matrix Equations ◽  
Quaternion Matrix ◽  
Common Solution ◽  
The Common

Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications

2009 ◽  
Vol 16 (02) ◽  
pp. 293-308 ◽  
Author(s):  
Qingwen Wang ◽  
Guangjing Song ◽  
Xin Liu
Keyword(s):  
Division Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Common Solution ◽  
Special Cases ◽  
The Common ◽  
Necessary And Sufficient ◽  
Linear Matrix Equations

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A1X = C1, XB2 = C2, A3XB3 = C3 and A4XB4 = C4 over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.

The Common Solution of Some Matrix Equations

2016 ◽  
Vol 23 (01) ◽  
pp. 71-81 ◽  
Author(s):  
Li Wang ◽  
Qingwen Wang ◽  
Zhuoheng He
Keyword(s):  
General Solution ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Existence Of A Solution ◽  
Solvability Conditions ◽  
Common Solution ◽  
The Common ◽  
Necessary And Sufficient ◽  
Linear Matrix Equations

In this paper we investigate the system of linear matrix equations A1X=C1, YB2=C2, A3XB3=C3, A4YB4=C4, BX+YC=A. We present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied.

scholarly journals Ranks of a Constrained Hermitian Matrix Expression with Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Shao-Wen Yu
Keyword(s):  
Hermitian Matrix ◽  
Schur Complement ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Necessary And Sufficient Condition ◽  
Common Solution ◽  
System Of Matrix Equations ◽  
Hermitian Solution ◽  
Matrix Expression ◽  
Necessary And Sufficient

We establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expressionC4−A4XA4∗whereXis a Hermitian solution to quaternion matrix equationsA1X=C1,XB1=C2, andA3XA3*=C3. As applications, we give a new necessary and sufficient condition for the existence of Hermitian solution to the system of matrix equationsA1X=C1,XB1=C2,A3XA3*=C3, andA4XA4*=C4, which was investigated by Wang and Wu, 2010, by rank equalities. In addition, extremal ranks of the generalized Hermitian Schur complementC4−A4A3~A4∗with respect to a Hermitian g-inverseA3~ofA3, which is a common solution to quaternion matrix equationsA1X=C1andXB1=C2, are also considered.

An easy way to create duration variables in binary cross-sectional time-series data

2020 ◽  
Vol 20 (4) ◽  
pp. 916-930
Author(s):  
Andrew Q. Philips
Keyword(s):  
Time Series ◽  
Missing Data ◽  
Time Series Data ◽  
Series Data ◽  
Duration Dependence ◽  
Cross Sectional ◽  
Common Solution ◽  
The Common

In cross-sectional time-series data with a dichotomous dependent variable, failing to account for duration dependence when it exists can lead to faulty inferences. A common solution is to include duration dummies, polynomials, or splines to proxy for duration dependence. Because creating these is not easy for the common practitioner, I introduce a new command, mkduration, that is a straightforward way to generate a duration variable for binary cross-sectional time-series data in Stata. mkduration can handle various forms of missing data and allows the duration variable to easily be turned into common parametric and nonparametric approximations.

scholarly journals A stochastic approach to handle resource constraints as knapsack problems in ensemble pruning

2021 ◽  
Author(s):  
András Hajdu ◽  
György Terdik ◽  
Attila Tiba ◽  
Henrietta Tomán
Keyword(s):  
Probability Function ◽  
Resource Constraints ◽  
Joint Probability ◽  
Stochastic Approach ◽  
Stochastic Search ◽  
Search Process ◽  
Common Solution ◽  
The Common ◽  
The Individual ◽  
Joint Probability Function

AbstractEnsemble-based methods are highly popular approaches that increase the accuracy of a decision by aggregating the opinions of individual voters. The common point is to maximize accuracy; however, a natural limitation occurs if incremental costs are also assigned to the individual voters. Consequently, we investigate creating ensembles under an additional constraint on the total cost of the members. This task can be formulated as a knapsack problem, where the energy is the ensemble accuracy formed by some aggregation rules. However, the generally applied aggregation rules lead to a nonseparable energy function, which takes the common solution tools—such as dynamic programming—out of action. We introduce a novel stochastic approach that considers the energy as the joint probability function of the member accuracies. This type of knowledge can be efficiently incorporated in a stochastic search process as a stopping rule, since we have the information on the expected accuracy or, alternatively, the probability of finding more accurate ensembles. Experimental analyses of the created ensembles of pattern classifiers and object detectors confirm the efficiency of our approach over other pruning ones. Moreover, we propose a novel stochastic search method that better fits the energy, which can be incorporated in other stochastic strategies as well.

scholarly journals A Parallel-Viscosity-Type Subgradient Extragradient-Line Method for Finding the Common Solution of Variational Inequality Problems Applied to Image Restoration Problems

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 248 ◽  
Author(s):  
Suthep Suantai ◽  
Pronpat Peeyada ◽  
Damrongsak Yambangwai ◽  
Watcharaporn Cholamjiak
Keyword(s):  
Variational Inequality ◽  
Hilbert Spaces ◽  
Hybrid Algorithm ◽  
Variational Inequality Problems ◽  
Strong Convergence Theorem ◽  
Infinite Dimensional ◽  
Common Solution ◽  
Line Method ◽  
The Common ◽  
Monotone Hybrid Algorithm

In this paper, we study a modified viscosity type subgradient extragradient-line method with a parallel monotone hybrid algorithm for approximating a common solution of variational inequality problems. Under suitable conditions in Hilbert spaces, the strong convergence theorem of the proposed algorithm to such a common solution is proved. We then give numerical examples in both finite and infinite dimensional spaces to justify our main theorem. Finally, we can show that our proposed algorithm is flexible and has good quality for use with common types of blur effects in image recovery.

An Iterative algorithm for $\eta$-(anti)-Hermitian least-squares solutions of quaternion matrix equations

2015 ◽  
Vol 30 ◽  
Author(s):  
Fatemeh Beik ◽  
Salman Ahmadi-Asl
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Iterative Algorithms ◽  
Iterative Approach ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Convergence Properties ◽  
Least Squares Problems

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

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