# American Institute of Mathematical Sciences

January  2015, 11(1): 171-183. doi: 10.3934/jimo.2015.11.171

## Optimization problems on the rank of the solution to left and right inverse eigenvalue problem

 1 School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, China, China 2 School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, Gansu 730030, China

Received  January 2013 Revised  December 2013 Published  May 2014

A complex matrix $P$ is called Hermitian and $\{k+1\}$-potent if $P^{k+1}=P=P^*$ for some integer $k\geq 1$. Let $P$ and $Q$ be $n\times n$ Hermitian and $\{k+1\}$-potent matrices, we say that complex matrix $A$ is $\{P,Q,k+1\}$-reflexive (anti-reflexive) if $PAQ=A$ ($PAQ=-A$). In this paper, the solvability conditions and the general solutions of the left and right inverse eigenvalue problem for $\{P,Q,k+1\}$-reflexive and anti-reflexive matrices are derived, and the minimal and maximal rank solutions are given. Moreover, the associated optimal approximation problem is also considered. Finally, numerical example is given to illustrate the main results.
Citation: Li-Fang Dai, Mao-Lin Liang, Wei-Yuan Ma. Optimization problems on the rank of the solution to left and right inverse eigenvalue problem. Journal of Industrial & Management Optimization, 2015, 11 (1) : 171-183. doi: 10.3934/jimo.2015.11.171
##### References:
 [1] A. Andrew, Solution of equations involving centrosymmetric matrices,, Technometrics, 15 (1973), 405. doi: 10.2307/1266998. [2] C. Beattie and S. Smith, Optimal matrix approximations in structural identification,, J. Optim. Theory Appl., 74 (1992), 23. doi: 10.1007/BF00939891. [3] P. Brussard and P. Glaudemans, Shell Model Applications in Nuclear Spectroscopy,, Elsevier, (1977). [4] H. Chen, Generalized reflexive matrices: Special properties and applications,, SIAN Matrix Anal. Appl., 19 (1998), 140. doi: 10.1137/S0895479895288759. [5] M. Chu and G. Golub, Inverse Eigenvalue Problems Theory, Algorithms, and Application,, Numerical Mathematics and Scientific Computation. Oxford University Press, (2005). doi: 10.1093/acprof:oso/9780198566649.001.0001. [6] B. N. Datta, Finite element model updating, eigenstructure assignment and eigenvalue embedding techniques for vibrating systems,, Mechanical Systems and Signal Processing, 16 (2002), 83. doi: 10.1006/mssp.2001.1443. [7] A. S. Deakin and T. M. Luke, On the inverse eigenvalue problems for matrices,, J. Phys. A, 25 (1992), 635. doi: 10.1088/0305-4470/25/3/020. [8] B. DeMoor and G. Golub, The restricted singular value decomposition: Properties and applcations,, SIAM J. Matrix Anal. Appl., 12 (1991), 401. doi: 10.1137/0612029. [9] A. Herrero and N. Thome, Using the GSVD and the lifting technique to find $\{P,k+1\}$-reflexive and anti-reflexive solutions of $AXB=C$., Appl. Math. Lett., 24 (2011), 1130. doi: 10.1016/j.aml.2011.01.039. [10] F. Li, X. Hu and L. Zhang, Left and right inverse eigenpairs problem of skew-centrosymmetric matrices,, Appl. Math. Comput., 177 (2006), 105. doi: 10.1016/j.amc.2005.10.035. [11] F. Li, X. Hu and L. Zhang, Left and right inverse eigenpairs problem of generalized centrosymmetric matrices and its optimal approximation problem,, Appl. Math. Comput., 212 (2009), 481. doi: 10.1016/j.amc.2009.02.035. [12] M. Liang and L. Dai, The left and right inverse eigenvalue problem for generalized reflexive and anti-reflexive matrices,, J. Comput. Appl., 234 (2010), 743. doi: 10.1016/j.cam.2010.01.014. [13] M. Liang, L. Dai and Y. Yang, The $\{P,Q, k+1\}$-reflexive solution of matrix equation $AXB= C$,, J. Appl. Math. Computing, 42 (2013), 339. doi: 10.1007/s12190-012-0631-3. [14] A. Marina, H. Daniel, M. Volkeer and C. Hans, The recursive inverse eigenvalue problem,, SIAM Matrix Anal. Appl., 22 (2000), 392. doi: 10.1137/S0895479899354044. [15] J. Paine, A numerical method for the inverse Sturm-Liouville problem,, SIAM J. Sci. Stat. Comput., 5 (1984), 149. doi: 10.1137/0905011. [16] H. Park, M. Jeon and J. Rosen, Low dimensional representation of text data in vector space based information retrievals,, Computat. Info. Retrieval, (2001), 3. [17] J. Respondek, Approximate controllability of the n-th Order infinite dimensional systems with controls delayed by the control devices,, Inter. Sys. Sci., 39 (2008), 765. doi: 10.1080/00207720701832655. [18] J. Respondek, On the confluent Vandermonde matrix calculation algorithm,, Appl. Math. Lett., 24 (2011), 103. doi: 10.1016/j.aml.2010.08.026. [19] J. Respondek, Numerical recipes for the high efficient inverse of the confluent, Vandermonde matrices,, Appl. Math. Comput., 218 (2011), 2044. doi: 10.1016/j.amc.2011.07.017. [20] J. Rosenthal and J. Willems, Open problems in the area of pole placement,, Open Problems in Mathematical Systems and Control Theory, (1999), 181. [21] Y. Tian, The maximal and minimal ranks of some expressions of generalized inverses of matrices,, Southeast Asian Bull. Math., 25 (2002), 745. doi: 10.1007/s100120200015. [22] Y. Tian and S. Cheng, The maximal and minimal ranks of $A-BXC$ with applications,, New York J. Math., 9 (2003), 345. [23] W. Trench, Minimization problems for (R,S)-symmetric and (R,S)-skew symmetric matrices,, Linear Algebra Appl., 389 (2004), 23. doi: 10.1016/j.laa.2004.03.035. [24] J. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors,, Am. Math. Mon., 92 (1985), 711. doi: 10.2307/2323222. [25] J. Wilkinson, The Algebraic Problem,, Oxford University Press, (1965). [26] D. Xie, X. Hu and Y. Sheng, The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations,, Linear Algebra Appl., 418 (2006), 142. doi: 10.1016/j.laa.2006.01.027. [27] L. Zadeh and C. Desoer, Linear System Theory: The State Space Approach,, McGraw Hill, (1963). [28] L. Zhang and D. Xie, A class of inverse eigenvalue problems,, Math. Sci. Acta, 13 (1993), 94.

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##### References:
 [1] A. Andrew, Solution of equations involving centrosymmetric matrices,, Technometrics, 15 (1973), 405. doi: 10.2307/1266998. [2] C. Beattie and S. Smith, Optimal matrix approximations in structural identification,, J. Optim. Theory Appl., 74 (1992), 23. doi: 10.1007/BF00939891. [3] P. Brussard and P. Glaudemans, Shell Model Applications in Nuclear Spectroscopy,, Elsevier, (1977). [4] H. Chen, Generalized reflexive matrices: Special properties and applications,, SIAN Matrix Anal. Appl., 19 (1998), 140. doi: 10.1137/S0895479895288759. [5] M. Chu and G. Golub, Inverse Eigenvalue Problems Theory, Algorithms, and Application,, Numerical Mathematics and Scientific Computation. Oxford University Press, (2005). doi: 10.1093/acprof:oso/9780198566649.001.0001. [6] B. N. Datta, Finite element model updating, eigenstructure assignment and eigenvalue embedding techniques for vibrating systems,, Mechanical Systems and Signal Processing, 16 (2002), 83. doi: 10.1006/mssp.2001.1443. [7] A. S. Deakin and T. M. Luke, On the inverse eigenvalue problems for matrices,, J. Phys. A, 25 (1992), 635. doi: 10.1088/0305-4470/25/3/020. [8] B. DeMoor and G. Golub, The restricted singular value decomposition: Properties and applcations,, SIAM J. Matrix Anal. Appl., 12 (1991), 401. doi: 10.1137/0612029. [9] A. Herrero and N. Thome, Using the GSVD and the lifting technique to find $\{P,k+1\}$-reflexive and anti-reflexive solutions of $AXB=C$., Appl. Math. Lett., 24 (2011), 1130. doi: 10.1016/j.aml.2011.01.039. [10] F. Li, X. Hu and L. Zhang, Left and right inverse eigenpairs problem of skew-centrosymmetric matrices,, Appl. Math. Comput., 177 (2006), 105. doi: 10.1016/j.amc.2005.10.035. [11] F. Li, X. Hu and L. Zhang, Left and right inverse eigenpairs problem of generalized centrosymmetric matrices and its optimal approximation problem,, Appl. Math. Comput., 212 (2009), 481. doi: 10.1016/j.amc.2009.02.035. [12] M. Liang and L. Dai, The left and right inverse eigenvalue problem for generalized reflexive and anti-reflexive matrices,, J. Comput. Appl., 234 (2010), 743. doi: 10.1016/j.cam.2010.01.014. [13] M. Liang, L. Dai and Y. Yang, The $\{P,Q, k+1\}$-reflexive solution of matrix equation $AXB= C$,, J. Appl. Math. Computing, 42 (2013), 339. doi: 10.1007/s12190-012-0631-3. [14] A. Marina, H. Daniel, M. Volkeer and C. Hans, The recursive inverse eigenvalue problem,, SIAM Matrix Anal. Appl., 22 (2000), 392. doi: 10.1137/S0895479899354044. [15] J. Paine, A numerical method for the inverse Sturm-Liouville problem,, SIAM J. Sci. Stat. Comput., 5 (1984), 149. doi: 10.1137/0905011. [16] H. Park, M. Jeon and J. Rosen, Low dimensional representation of text data in vector space based information retrievals,, Computat. Info. Retrieval, (2001), 3. [17] J. Respondek, Approximate controllability of the n-th Order infinite dimensional systems with controls delayed by the control devices,, Inter. Sys. Sci., 39 (2008), 765. doi: 10.1080/00207720701832655. [18] J. Respondek, On the confluent Vandermonde matrix calculation algorithm,, Appl. Math. Lett., 24 (2011), 103. doi: 10.1016/j.aml.2010.08.026. [19] J. Respondek, Numerical recipes for the high efficient inverse of the confluent, Vandermonde matrices,, Appl. Math. Comput., 218 (2011), 2044. doi: 10.1016/j.amc.2011.07.017. [20] J. Rosenthal and J. Willems, Open problems in the area of pole placement,, Open Problems in Mathematical Systems and Control Theory, (1999), 181. [21] Y. Tian, The maximal and minimal ranks of some expressions of generalized inverses of matrices,, Southeast Asian Bull. Math., 25 (2002), 745. doi: 10.1007/s100120200015. [22] Y. Tian and S. Cheng, The maximal and minimal ranks of $A-BXC$ with applications,, New York J. Math., 9 (2003), 345. [23] W. Trench, Minimization problems for (R,S)-symmetric and (R,S)-skew symmetric matrices,, Linear Algebra Appl., 389 (2004), 23. doi: 10.1016/j.laa.2004.03.035. [24] J. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors,, Am. Math. Mon., 92 (1985), 711. doi: 10.2307/2323222. [25] J. Wilkinson, The Algebraic Problem,, Oxford University Press, (1965). [26] D. Xie, X. Hu and Y. Sheng, The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations,, Linear Algebra Appl., 418 (2006), 142. doi: 10.1016/j.laa.2006.01.027. [27] L. Zadeh and C. Desoer, Linear System Theory: The State Space Approach,, McGraw Hill, (1963). [28] L. Zhang and D. Xie, A class of inverse eigenvalue problems,, Math. Sci. Acta, 13 (1993), 94.
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