On the Kronecker Products and Their Applications

This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. In addition, we establish the relations between the singular values of two matrices and their Kronecker product and the relations between the determinant, the trace, the rank, and the polynomial matrix of the Kronecker products.

[1]  Qin-Zhen Huang.  Consensus Analysis of Multi-agent Discrete-time Systems , 2012 .

[2]  P. Lancaster,et al.  The theory of matrices : with applications , 1985 .

[3]  Ai-Guo Wu,et al.  Iterative solutions to the extended Sylvester-conjugate matrix equations , 2010, Appl. Math. Comput..

[4]  Huazhen Fang,et al.  Kalman filter‐based adaptive control for networked systems with unknown parameters and randomly missing outputs , 2009 .

[5]  D. Bahuguna,et al.  Advanced type coupled matrix Riccati differential equation systems with Kronecker product , 2007, Appl. Math. Comput..

[6]  Y. Liu,et al.  Gradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systems , 2012, J. Syst. Control. Eng..

[7]  J. Chu,et al.  Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle , 2013 .

[8]  W. Steeb,et al.  Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra , 2011 .

[9]  Marko Huhtanen.  Real linear Kronecker product operations , 2006 .

[10]  C. Loan.  The ubiquitous Kronecker product , 2000 .

[11]  Feng Ding,et al.  Transformations between some special matrices , 2010, Comput. Math. Appl..

[12]  H. Neudecker,et al.  Block Kronecker products and the vecb operator , 1991 .

[13]  F. Ding,et al.  Multistage least squares based iterative estimation for feedback nonlinear systems with moving average noises using the hierarchical identification principle , 2013 .

[14]  Jie Ding,et al.  Bias compensation‐based parameter estimation for output error moving average systems , 2011 .

[15]  Feng Ding,et al.  Iterative least-squares solutions of coupled Sylvester matrix equations , 2005, Syst. Control. Lett..

[16]  Feng Ding,et al.  Performance analysis of the auxiliary model-based least-squares identification algorithm for one-step state-delay systems , 2012, Int. J. Comput. Math..

[17]  Feng Ding,et al.  Hierarchical Least Squares Identification for Linear SISO Systems With Dual-Rate Sampled-Data , 2011, IEEE Transactions on Automatic Control.

[18]  J. Magnus,et al.  Matrix differential calculus with applications to simple, Hadamard, and Kronecker products. , 1985 .

[19]  Jie Ding,et al.  Time series AR modeling with missing observations based on the polynomial transformation , 2010, Math. Comput. Model..

[20]  W.-H Steeba,et al.  Exponential functions of Kronecker products and trace calculation , 1981 .

[21]  J. Turner,et al.  Qualitative properties of adjoint Kronecker product boundary value problems , 2002, Appl. Math. Comput..

[22]  Feng Ding,et al.  Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle , 2008, Appl. Math. Comput..

[23]  Mehdi Dehghan,et al.  An efficient algorithm for solving general coupled matrix equations and its application , 2010, Math. Comput. Model..

[24]  Mehdi Dehghan,et al.  An iterative algorithm for solving a pair of matrix equations AYB=E, CYD=F over generalized centro-symmetric matrices , 2008, Comput. Math. Appl..

[25]  Xiangpeng Xie,et al.  Relaxed Stability Conditions for Continuous-Time T–S Fuzzy-Control Systems Via Augmented Multi-Indexed Matrix Approach , 2011, IEEE Transactions on Fuzzy Systems.

[26]  Yang Shi,et al.  Output Feedback Stabilization of Networked Control Systems With Random Delays Modeled by Markov Chains , 2009, IEEE Transactions on Automatic Control.

[27]  Alexander Graham,et al.  Kronecker Products and Matrix Calculus: With Applications , 1981 .

[28]  Feng Ding,et al.  Two-stage parameter estimation algorithms for Box-Jenkins systems , 2013, IET Signal Process..

[29]  P. Bentler,et al.  Matrix derivatives with chain rule and rules for simple, Hadamard, and Kronecker products , 1978 .

[30]  Mehdi Dehghan,et al.  An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation , 2008, Appl. Math. Comput..

[31]  S. Delvaux,et al.  Rank-deficient submatrices of Kronecker products of Fourier matrices , 2007 .

[32]  Ruifeng Ding,et al.  Iterative parameter identification methods for nonlinear functions , 2012 .

[33]  Ai-Guo Wu,et al.  Iterative algorithms for solving a class of complex conjugate and transpose matrix equations , 2011, Appl. Math. Comput..

[34]  F. Ding.  Two-stage least squares based iterative estimation algorithm for CARARMA system modeling ☆ , 2013 .

[35]  Yanjun Liu,et al.  Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model , 2009, Appl. Math. Comput..

[36]  F. Ding.  Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling , 2013 .

[37]  Fengxia Zhang,et al.  Least squares solutions with special structure to the linear matrix equation AXB = C , 2011, Appl. Math. Comput..

[38]  Yu-Fan Zheng,et al.  Dynamic Behavior of Multi-agent Systems with Distributed Sampled Control: Dynamic Behavior of Multi-agent Systems with Distributed Sampled Control , 2012 .

[39]  Feng Ding,et al.  Iterative solutions to matrix equations of the form AiXBi=Fi , 2010, Comput. Math. Appl..

[40]  Feng Ding,et al.  Performance Analysis of the Auxiliary Model-Based Stochastic Gradient Parameter Estimation Algorithm for State-Space Systems with One-Step State Delay , 2013, Circuits Syst. Signal Process..

[41]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[42]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[43]  Weili Xiong,et al.  Least-Squares Parameter Estimation Algorithm for a Class of Input Nonlinear Systems , 2012, J. Appl. Math..

[44]  F. Ding.  Coupled-least-squares identification for multivariable systems , 2013 .

[45]  Bing Chen,et al.  Synchronization for Coupled Neural Networks With Interval Delay: A Novel Augmented Lyapunov–Krasovskii Functional Method , 2013, IEEE Transactions on Neural Networks and Learning Systems.

[46]  H. Abou-Kandil,et al.  Kronecker products and coupled matrix Riccati differential systems , 1989 .

[47]  Yanjun Liu,et al.  Gradient based and least squares based iterative algorithms for matrix equations AXB + CXTD = F , 2010, Appl. Math. Comput..

[48]  Yongge Tian,et al.  Rank equalities and inequalities for Kronecker products of matrices with applications , 2004, Appl. Math. Comput..

[49]  Ali Kaveh,et al.  Eigensolution of Laplacian matrices for graph partitioning and domain decomposition , 2009 .

[50]  Feng Ding,et al.  On Iterative Solutions of General Coupled Matrix Equations , 2006, SIAM J. Control. Optim..

[51]  Jie Sheng,et al.  Convergence of stochastic gradient estimation algorithm for multivariable ARX-like systems , 2010, Comput. Math. Appl..

[52]  Mehdi Dehghan,et al.  SSHI methods for solving general linear matrix equations , 2011 .

[53]  Feng Ding,et al.  Decomposition based fast least squares algorithm for output error systems , 2013, Signal Process..

[54]  Junhong Li,et al.  Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration , 2013, Appl. Math. Lett..

[55]  Feng Ding,et al.  Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations , 2005, IEEE Trans. Autom. Control..

[56]  Mehdi Dehghan,et al.  An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices , 2010 .

[57]  S. R. Searle,et al.  On the history of the kronecker product , 1983 .

[58]  M. A. Bradford,et al.  Coupling of finite element and meshfree methods for locking‐free analysis of shear‐deformable beams and plates , 2011 .