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Transformations between some special matrices. (English) Zbl 1193.15028

Summary: Special matrices are very useful in signal processing and control systems. This paper studies the transformations and relationships between some special matrices. The conditions that a matrix is similar to a companion matrix are derived. It is proved that a companion matrix is similar to a diagonal matrix or Jordan matrix, and the transformation matrices between them are given. Finally, we apply the similarity transformation and the companion matrix to system identification.

MSC:

15B34 Boolean and Hadamard matrices
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[1] Al Zhour, Z.; Kilicman, A., Some new connections between matrix products for partitioned and non-partitioned matrices, Computers & Mathematics with Applications, 54, 6, 763-784 (2007) · Zbl 1146.15014
[2] Ding, F.; Chen, T., Iterative least squares solutions of coupled Sylvester matrix equations, Systems & Control Letters, 54, 2, 95-107 (2005) · Zbl 1129.65306
[3] Ding, F.; Chen, T., On iterative solutions of general coupled matrix equations, SIAM Journal on Control and Optimization, 44, 6, 2269-2284 (2006) · Zbl 1115.65035
[4] Kilicman, A.; Al Zhour, Z., Vector least-squares solutions for coupled singular matrix equations, Journal of Computational and Applied Mathematics, 206, 2, 1051-1069 (2007) · Zbl 1132.65034
[5] Dehghan, M.; Hajarian, M., An iterative algorithm for solving a pair of matrix equations \(A Y B = E, C Y D = F\) over generalized centro-symmetric matrices, Computers & Mathematics with Applications, 56, 12, 3246-3260 (2008) · Zbl 1165.15301
[6] Dehghan, M.; Hajarian, M., Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation \(A_1 X_1 B_1 + A_2 X_2 B_2 = C\), Mathematical and Computer Modelling, 49, 9-10, 1937-1959 (2009) · Zbl 1171.15310
[7] Dehghan, M.; Hajarian, M., A lower bound for the product of eigenvalues of solutions to matrix equations, Applied Mathematics Letters, 22, 12, 1786-1788 (2009) · Zbl 1190.15022
[8] Ding, F.; Chen, T., Gradient based iterative algorithms for solving a class of matrix equations, IEEE Transactions on Automatic Control, 50, 8, 1216-1221 (2005) · Zbl 1365.65083
[9] Mukaidani, H.; Yamamoto, S.; Yamamoto, T., A numerical algorithm for finding solution of cross-coupled algebraic Riccati equations, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, E91A, 2, 682-685 (2008)
[10] Zhou, B.; Duan, G. R.; Li, Z. Y., A Stein matrix equation approach for computing coprime matrix fraction description, IET Control Theory and Applications, 3, 6, 691-700 (2009)
[11] Zhou, B.; Li, Z. Y.; Duan, G. R., Solutions to a family of matrix equations by using the Kronecker matrix polynomials, Applied Mathematics and Computation, 212, 2, 327-336 (2009) · Zbl 1181.15020
[12] Ding, F.; Liu, P. X.; Ding, J., Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Applied Mathematics and Computation, 197, 1, 41-50 (2008) · Zbl 1143.65035
[13] Xie, L.; Ding, J.; Ding, F., Gradient based iterative solutions for general linear matrix equations, Computers & Mathematics with Applications, 58, 7, 1441-1448 (2009) · Zbl 1189.65083
[14] Ding, F.; Chen, T., Hierarchical gradient based identification of multivariable discrete-time systems, Automatica, 41, 2, 315-325 (2005) · Zbl 1073.93012
[15] Ding, F.; Chen, T., Hierarchical least squares identification methods for multivariable systems, IEEE Transactions on Automatic Control, 50, 3, 397-402 (2005) · Zbl 1365.93551
[16] Ding, F.; Chen, T., Hierarchical identification of lifted state-space models for general dual-rate systems, IEEE Transactions on Circuits and Systems, 52, 6, 1179-1187 (2005) · Zbl 1374.93342
[17] Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), Johns Hopkins Univ. Press: Johns Hopkins Univ. Press Baltimore, MD · Zbl 0865.65009
[18] Hou, S. H.; Pang, W. K., Inversion of confluent Vandermonde matrices, Computers & Mathematics with Applications, 43, 12, 1539-1547 (2002) · Zbl 1002.65036
[19] Ding, F.; Chen, T., Performance analysis of multi-innovation gradient type identification methods, Automatica, 43, 1, 1-14 (2007) · Zbl 1140.93488
[20] Ding, F.; Liu, P. X.; Yang, H. Z., Parameter identification and intersample output estimation for dual-rate systems, IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans, 38, 4, 966-975 (2008)
[21] Ding, F.; Liu, P. X.; Liu, G., Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises, Signal Processing, 89, 10, 1883-1890 (2009) · Zbl 1178.94137
[22] Ding, F.; Qiu, L.; Chen, T., Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems, Automatica, 45, 2, 324-332 (2009) · Zbl 1158.93365
[23] Han, L. L.; Ding, F., Multi-innovation stochastic gradient algorithms for multi-input multi-output systems, Digital Signal Processing, 19, 4, 545-554 (2009)
[24] Han, L. L.; Ding, F., Identification for multirate multi-input systems using the multi-innovation identification theory, Computers & Mathematics with Applications, 57, 9, 1438-1449 (2009) · Zbl 1186.93076
[25] Ding, J.; Ding, F., The residual based extended least squares identification method for dual-rate systems, Computers & Mathematics with Applications, 56, 6, 1479-1487 (2008) · Zbl 1155.93435
[26] Wang, D. Q.; Ding, F., Extended stochastic gradient identification algorithms for Hammerstein-Wiener ARMAX systems, Computers & Mathematics with Applications, 56, 12, 3157-3164 (2008) · Zbl 1165.65308
[27] Liu, Y. J.; Xiao, Y. S.; Zhao, X. L., Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model, Applied Mathematics and Computation, 215, 4, 1477-1483 (2009) · Zbl 1177.65095
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