Efficient Solutions of Coupled Matrix and Matrix Differential Equations

Author(s)
Zeyad Al-Zhour

Affiliation(s)

Department of Basic Sciences and Humanities, College of Engineering, University of Dammam, Dammam, Kingdom of Saudi Arabia.

Department of Basic Sciences and Humanities, College of Engineering, University of Dammam, Dammam, Kingdom of Saudi Arabia.

ABSTRACT

In Kronecker products works, matrices are sometimes regarded as vectors and vectors are sometimes made in to matrices. To be precise about these reshaping we use the vector and diagonal extraction operators. In the present paper, the results are organized in the following ways. First, we formulate the coupled matrix linear least-squares problem and present the efficient solutions of this problem that arises in multistatic antenna array processing problem. Second, we extend the use of connection between the Hadamard (Kronecker) product and diagonal extraction (vector) operator in order to construct a computationally-efficient solution of non-homogeneous coupled matrix differential equations that useful in various applications. Finally, the analysis indicates that the Kronecker (Khatri-Rao) structure method can achieve good efficient while the Hadamard structure method achieve more efficient when the unknown matrices are diagonal.

In Kronecker products works, matrices are sometimes regarded as vectors and vectors are sometimes made in to matrices. To be precise about these reshaping we use the vector and diagonal extraction operators. In the present paper, the results are organized in the following ways. First, we formulate the coupled matrix linear least-squares problem and present the efficient solutions of this problem that arises in multistatic antenna array processing problem. Second, we extend the use of connection between the Hadamard (Kronecker) product and diagonal extraction (vector) operator in order to construct a computationally-efficient solution of non-homogeneous coupled matrix differential equations that useful in various applications. Finally, the analysis indicates that the Kronecker (Khatri-Rao) structure method can achieve good efficient while the Hadamard structure method achieve more efficient when the unknown matrices are diagonal.

Cite this paper

Z. Al-Zhour, "Efficient Solutions of Coupled Matrix and Matrix Differential Equations,"*Intelligent Control and Automation*, Vol. 3 No. 2, 2012, pp. 176-187. doi: 10.4236/ica.2012.32020.

Z. Al-Zhour, "Efficient Solutions of Coupled Matrix and Matrix Differential Equations,"

References

[1] T. Kailath, “Linear Systems,” Prentice-Hall, Englewood Cliffs, 1980.

[2] T. Kailath and A. H. Sayed, “Displacement Structure: Theory and Applications,” SIAM Review, Vol. 37, No. 3, 1995, pp. 297-386. doi:10.1137/1037082

[3] G. Mouroutsos and P. D Sparis, “Taylor Series Approach to System Identification, Analysis and Optimal Control,” Journal of the Franklin Institute, Vol. 319, No. 3, pp. 359-371. doi:10.1016/0016-0032(85)90056-0

[4] L. Jódar and H. Abou-Kandil, “Kronecker Products and Coupled Matrix Riccati Differential Systems,” Linear Algebra and its Applications, Vol. 121, 1989, pp. 39-51. doi:10.1016/0024-3795(89)90690-3

[5] S. Campbell, “Singular Systems of Differential Equations II,” Pitman, London, 1982.

[6] M. Mariton, “Les Systèmes Linéaires á Sauts Markoviens,” Thèse d’Etat, Université Paris-Sud, 1986.

[7] J. B. Cruz and C. I. Chen, “Series Nash Solution of TwoPerson Nonzero Sum Linear Differential Games,” Journal of Optimization Theory and Applications, Vol. 7, No. 4, 1971, pp. 240-257. doi:10.1007/BF00928706

[8] Z. Al-Zhour and A. Kilicman, “Matrix Equalities and Inequalities Involving Khatri-Rao and Tracy-Singh Sums,” Journal of Inequalities in Pure & Applied Mathematics, Vol. 7, No. 1, 2006, pp. 496-513.

[9] A. Graham, “Kronecker Products and Matrix Calculus with Applications,” Ellis Horwood Ltd., New York, 1981.

[10] A. Kilicman and Z. Al-Zhour, “Vector Least-Squares Solutions of Coupled Singular Matrix Equations,” Journal of Computational and Applied Mathematics, Vol. 206, No. 2, 2007, pp. 1051-1069. doi:10.1016/j.cam.2006.09.009

[11] W.-H. Steeb, “Matrix Calculus and Kronecker Product with Applications and C++ Programs,” World Scientific Publishing Co. Pte. Ltd., Singapore, 1997.

[12] G. Visick, “A Quantitative Version of the Observation that the Hadamard Product Is A Principle Submatrix of the Kronecker Product,” Linear Algebra and its Applications, Vol. 304, No. 1-3, 2000, pp. 45-68. doi:10.1016/S0024-3795(99)00187-1

[13] H. Lev-Ari, “Efficient Solution of Linear Matrix Equations with Application to Multistatic Antenna Array Processing,” Communications in Information & Systems, Vol. 5, No. 1, 2005, pp. 123-130.

[14] C. R. Rao and M. B. Rao, “Matrix Algebra and its Applications to Statistics and Econometrics,” World Scientific Publishing Co. Pte. Ltd., Singapore, 1998.

[15] F. Ding and T. Chen, “Iterative Least-Squares Solutions of Coupled Sylvester Matrix Equations,” Systems & Control Letters, Vol. 54, No. 2, 2005, pp. 95-107. doi:10.1016/j.sysconle.2004.06.008

[16] S. Tauber, “An Applications of the Hadamard Product to Air Pollution,” Applied Mathematics and Computation, Vol. 4, No. 2, 1978, pp. 167-176. doi:10.1016/0096-3003(78)90020-6

[17] F. Zhang, Matrix Theory: Basic Results and Techniques, Springer-Verlag, New York, 1999.

[18] A. Kilicman and Z. Al-Zhour, “The General Common Exact Solutions of Coupled Linear Matrix and Matrix Differential Equations,” Journal of Computational Analysis and Applications, Vol. 1, No. 1, 2005, pp. 15-29.

[19] J. R. Magnus and H. Neudecker, “Matrix Differential Calculus with Applications in Statistics and Econometrics,” John Wiley and Sons Ltd., New York, 1999.

[20] G. F. Van Loan, “The Ubiquitous Kronecker Product,” Journal of Computational and Applied Mathematics, Vol. 123, No. 1-2, 2000, pp. 85-100. doi:10.1016/S0377-0427(00)00393-9

[1] T. Kailath, “Linear Systems,” Prentice-Hall, Englewood Cliffs, 1980.

[2] T. Kailath and A. H. Sayed, “Displacement Structure: Theory and Applications,” SIAM Review, Vol. 37, No. 3, 1995, pp. 297-386. doi:10.1137/1037082

[3] G. Mouroutsos and P. D Sparis, “Taylor Series Approach to System Identification, Analysis and Optimal Control,” Journal of the Franklin Institute, Vol. 319, No. 3, pp. 359-371. doi:10.1016/0016-0032(85)90056-0

[4] L. Jódar and H. Abou-Kandil, “Kronecker Products and Coupled Matrix Riccati Differential Systems,” Linear Algebra and its Applications, Vol. 121, 1989, pp. 39-51. doi:10.1016/0024-3795(89)90690-3

[5] S. Campbell, “Singular Systems of Differential Equations II,” Pitman, London, 1982.

[6] M. Mariton, “Les Systèmes Linéaires á Sauts Markoviens,” Thèse d’Etat, Université Paris-Sud, 1986.

[7] J. B. Cruz and C. I. Chen, “Series Nash Solution of TwoPerson Nonzero Sum Linear Differential Games,” Journal of Optimization Theory and Applications, Vol. 7, No. 4, 1971, pp. 240-257. doi:10.1007/BF00928706

[8] Z. Al-Zhour and A. Kilicman, “Matrix Equalities and Inequalities Involving Khatri-Rao and Tracy-Singh Sums,” Journal of Inequalities in Pure & Applied Mathematics, Vol. 7, No. 1, 2006, pp. 496-513.

[9] A. Graham, “Kronecker Products and Matrix Calculus with Applications,” Ellis Horwood Ltd., New York, 1981.

[10] A. Kilicman and Z. Al-Zhour, “Vector Least-Squares Solutions of Coupled Singular Matrix Equations,” Journal of Computational and Applied Mathematics, Vol. 206, No. 2, 2007, pp. 1051-1069. doi:10.1016/j.cam.2006.09.009

[11] W.-H. Steeb, “Matrix Calculus and Kronecker Product with Applications and C++ Programs,” World Scientific Publishing Co. Pte. Ltd., Singapore, 1997.

[12] G. Visick, “A Quantitative Version of the Observation that the Hadamard Product Is A Principle Submatrix of the Kronecker Product,” Linear Algebra and its Applications, Vol. 304, No. 1-3, 2000, pp. 45-68. doi:10.1016/S0024-3795(99)00187-1

[13] H. Lev-Ari, “Efficient Solution of Linear Matrix Equations with Application to Multistatic Antenna Array Processing,” Communications in Information & Systems, Vol. 5, No. 1, 2005, pp. 123-130.

[14] C. R. Rao and M. B. Rao, “Matrix Algebra and its Applications to Statistics and Econometrics,” World Scientific Publishing Co. Pte. Ltd., Singapore, 1998.

[15] F. Ding and T. Chen, “Iterative Least-Squares Solutions of Coupled Sylvester Matrix Equations,” Systems & Control Letters, Vol. 54, No. 2, 2005, pp. 95-107. doi:10.1016/j.sysconle.2004.06.008

[16] S. Tauber, “An Applications of the Hadamard Product to Air Pollution,” Applied Mathematics and Computation, Vol. 4, No. 2, 1978, pp. 167-176. doi:10.1016/0096-3003(78)90020-6

[17] F. Zhang, Matrix Theory: Basic Results and Techniques, Springer-Verlag, New York, 1999.

[18] A. Kilicman and Z. Al-Zhour, “The General Common Exact Solutions of Coupled Linear Matrix and Matrix Differential Equations,” Journal of Computational Analysis and Applications, Vol. 1, No. 1, 2005, pp. 15-29.

[19] J. R. Magnus and H. Neudecker, “Matrix Differential Calculus with Applications in Statistics and Econometrics,” John Wiley and Sons Ltd., New York, 1999.

[20] G. F. Van Loan, “The Ubiquitous Kronecker Product,” Journal of Computational and Applied Mathematics, Vol. 123, No. 1-2, 2000, pp. 85-100. doi:10.1016/S0377-0427(00)00393-9