# 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. Google Scholar [2] C. Beattie and S. Smith, Optimal matrix approximations in structural identification,, J. Optim. Theory Appl., 74 (1992), 23. doi: 10.1007/BF00939891. Google Scholar [3] P. Brussard and P. Glaudemans, Shell Model Applications in Nuclear Spectroscopy,, Elsevier, (1977). Google Scholar [4] H. Chen, Generalized reflexive matrices: Special properties and applications,, SIAN Matrix Anal. Appl., 19 (1998), 140. doi: 10.1137/S0895479895288759. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] J. Paine, A numerical method for the inverse Sturm-Liouville problem,, SIAM J. Sci. Stat. Comput., 5 (1984), 149. doi: 10.1137/0905011. Google Scholar [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. Google Scholar [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. Google Scholar [18] J. Respondek, On the confluent Vandermonde matrix calculation algorithm,, Appl. Math. Lett., 24 (2011), 103. doi: 10.1016/j.aml.2010.08.026. Google Scholar [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. Google Scholar [20] J. Rosenthal and J. Willems, Open problems in the area of pole placement,, Open Problems in Mathematical Systems and Control Theory, (1999), 181. Google Scholar [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. Google Scholar [22] Y. Tian and S. Cheng, The maximal and minimal ranks of $A-BXC$ with applications,, New York J. Math., 9 (2003), 345. Google Scholar [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. Google Scholar [24] J. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors,, Am. Math. Mon., 92 (1985), 711. doi: 10.2307/2323222. Google Scholar [25] J. Wilkinson, The Algebraic Problem,, Oxford University Press, (1965). Google Scholar [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. Google Scholar [27] L. Zadeh and C. Desoer, Linear System Theory: The State Space Approach,, McGraw Hill, (1963). Google Scholar [28] L. Zhang and D. Xie, A class of inverse eigenvalue problems,, Math. Sci. Acta, 13 (1993), 94. Google Scholar

show all references

##### References:
 [1] A. Andrew, Solution of equations involving centrosymmetric matrices,, Technometrics, 15 (1973), 405. doi: 10.2307/1266998. Google Scholar [2] C. Beattie and S. Smith, Optimal matrix approximations in structural identification,, J. Optim. Theory Appl., 74 (1992), 23. doi: 10.1007/BF00939891. Google Scholar [3] P. Brussard and P. Glaudemans, Shell Model Applications in Nuclear Spectroscopy,, Elsevier, (1977). Google Scholar [4] H. Chen, Generalized reflexive matrices: Special properties and applications,, SIAN Matrix Anal. Appl., 19 (1998), 140. doi: 10.1137/S0895479895288759. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] J. Paine, A numerical method for the inverse Sturm-Liouville problem,, SIAM J. Sci. Stat. Comput., 5 (1984), 149. doi: 10.1137/0905011. Google Scholar [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. Google Scholar [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. Google Scholar [18] J. Respondek, On the confluent Vandermonde matrix calculation algorithm,, Appl. Math. Lett., 24 (2011), 103. doi: 10.1016/j.aml.2010.08.026. Google Scholar [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. Google Scholar [20] J. Rosenthal and J. Willems, Open problems in the area of pole placement,, Open Problems in Mathematical Systems and Control Theory, (1999), 181. Google Scholar [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. Google Scholar [22] Y. Tian and S. Cheng, The maximal and minimal ranks of $A-BXC$ with applications,, New York J. Math., 9 (2003), 345. Google Scholar [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. Google Scholar [24] J. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors,, Am. Math. Mon., 92 (1985), 711. doi: 10.2307/2323222. Google Scholar [25] J. Wilkinson, The Algebraic Problem,, Oxford University Press, (1965). Google Scholar [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. Google Scholar [27] L. Zadeh and C. Desoer, Linear System Theory: The State Space Approach,, McGraw Hill, (1963). Google Scholar [28] L. Zhang and D. Xie, A class of inverse eigenvalue problems,, Math. Sci. Acta, 13 (1993), 94. Google Scholar
 [1] Haixia Liu, Jian-Feng Cai, Yang Wang. Subspace clustering by (k,k)-sparse matrix factorization. Inverse Problems & Imaging, 2017, 11 (3) : 539-551. doi: 10.3934/ipi.2017025 [2] Hsin-Yi Liu, Hsing Paul Luh. Kronecker product-forms of steady-state probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 691-711. doi: 10.3934/naco.2011.1.691 [3] Yuhua Sun, Zilong Wang, Hui Li, Tongjiang Yan. The cross-correlation distribution of a $p$-ary $m$-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$. Advances in Mathematics of Communications, 2013, 7 (4) : 409-424. doi: 10.3934/amc.2013.7.409 [4] Roberta Ghezzi, Frédéric Jean. A new class of $(H^k,1)$-rectifiable subsets of metric spaces. Communications on Pure & Applied Analysis, 2013, 12 (2) : 881-898. doi: 10.3934/cpaa.2013.12.881 [5] Gaidi Li, Zhen Wang, Dachuan Xu. An approximation algorithm for the $k$-level facility location problem with submodular penalties. Journal of Industrial & Management Optimization, 2012, 8 (3) : 521-529. doi: 10.3934/jimo.2012.8.521 [6] Marek Galewski, Renata Wieteska. Multiple periodic solutions to a discrete $p^{(k)}$ - Laplacian problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2535-2547. doi: 10.3934/dcdsb.2014.19.2535 [7] Zhengshan Dong, Jianli Chen, Wenxing Zhu. Homotopy method for matrix rank minimization based on the matrix hard thresholding method. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 211-224. doi: 10.3934/naco.2019015 [8] Weigao Ge, Li Zhang. Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4925-4943. doi: 10.3934/dcds.2016013 [9] Jacek Banasiak, Amartya Goswami. Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 617-635. doi: 10.3934/dcds.2015.35.617 [10] Hua Liang, Jinquan Luo, Yuansheng Tang. On cross-correlation of a binary $m$-sequence of period $2^{2k}-1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$. Advances in Mathematics of Communications, 2017, 11 (4) : 693-703. doi: 10.3934/amc.2017050 [11] Feng Rong. Non-algebraic attractors on $\mathbf{P}^k$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 977-989. doi: 10.3934/dcds.2012.32.977 [12] Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 [13] Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems & Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601 [14] Yongge Tian. A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 289-326. doi: 10.3934/naco.2015.5.289 [15] Xianchao Xiu, Lingchen Kong. Rank-one and sparse matrix decomposition for dynamic MRI. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 127-134. doi: 10.3934/naco.2015.5.127 [16] Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016 [17] Jing Zhou, Dejun Chen, Zhenbo Wang, Wenxun Xing. A conic approximation method for the 0-1 quadratic knapsack problem. Journal of Industrial & Management Optimization, 2013, 9 (3) : 531-547. doi: 10.3934/jimo.2013.9.531 [18] Roberto Triggiani. A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: A critical subspace of $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^*)^{\frac{1}{2}})$ and implications. Evolution Equations & Control Theory, 2016, 5 (1) : 185-199. doi: 10.3934/eect.2016.5.185 [19] Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791 [20] Jussi Korpela, Matti Lassas, Lauri Oksanen. Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation. Inverse Problems & Imaging, 2019, 13 (3) : 575-596. doi: 10.3934/ipi.2019027

2018 Impact Factor: 1.025