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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 |
References:
[1] |
A. Andrew, Solution of equations involving centrosymmetric matrices, Technometrics, 15 (1973), 405-407.
doi: 10.2307/1266998. |
[2] |
C. Beattie and S. Smith, Optimal matrix approximations in structural identification, J. Optim. Theory Appl., 74 (1992), 23-56.
doi: 10.1007/BF00939891. |
[3] |
P. Brussard and P. Glaudemans, Shell Model Applications in Nuclear Spectroscopy, Elsevier, New York, 1977. |
[4] |
H. Chen, Generalized reflexive matrices: Special properties and applications, SIAN Matrix Anal. Appl., 19 (1998), 140-153.
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, New York, 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), 83-96.
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), 635-648.
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), 401-425.
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), 1130-1141.
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), 105-110.
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), 481-487.
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), 743-749.
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), 339-350.
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), 392-412.
doi: 10.1137/S0895479899354044. |
[15] |
J. Paine, A numerical method for the inverse Sturm-Liouville problem, SIAM J. Sci. Stat. Comput., 5 (1984), 149-156.
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), 3-23. |
[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), 765-782.
doi: 10.1080/00207720701832655. |
[18] |
J. Respondek, On the confluent Vandermonde matrix calculation algorithm, Appl. Math. Lett., 24 (2011), 103-106.
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), 2044-2054.
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, Springer, London, (1999), 181-191. |
[21] |
Y. Tian, The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math., 25 (2002), 745-755.
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), 345-362. |
[23] |
W. Trench, Minimization problems for (R,S)-symmetric and (R,S)-skew symmetric matrices, Linear Algebra Appl., 389 (2004), 23-31.
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), 711-717.
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), 142-152.
doi: 10.1016/j.laa.2006.01.027. |
[27] |
L. Zadeh and C. Desoer, Linear System Theory: The State Space Approach, McGraw Hill, New York, 1963. |
[28] |
L. Zhang and D. Xie, A class of inverse eigenvalue problems, Math. Sci. Acta, 13 (1993), 94-99. |
show all references
References:
[1] |
A. Andrew, Solution of equations involving centrosymmetric matrices, Technometrics, 15 (1973), 405-407.
doi: 10.2307/1266998. |
[2] |
C. Beattie and S. Smith, Optimal matrix approximations in structural identification, J. Optim. Theory Appl., 74 (1992), 23-56.
doi: 10.1007/BF00939891. |
[3] |
P. Brussard and P. Glaudemans, Shell Model Applications in Nuclear Spectroscopy, Elsevier, New York, 1977. |
[4] |
H. Chen, Generalized reflexive matrices: Special properties and applications, SIAN Matrix Anal. Appl., 19 (1998), 140-153.
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, New York, 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), 83-96.
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), 635-648.
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), 401-425.
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), 1130-1141.
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), 105-110.
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), 481-487.
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), 743-749.
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), 339-350.
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), 392-412.
doi: 10.1137/S0895479899354044. |
[15] |
J. Paine, A numerical method for the inverse Sturm-Liouville problem, SIAM J. Sci. Stat. Comput., 5 (1984), 149-156.
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), 3-23. |
[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), 765-782.
doi: 10.1080/00207720701832655. |
[18] |
J. Respondek, On the confluent Vandermonde matrix calculation algorithm, Appl. Math. Lett., 24 (2011), 103-106.
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), 2044-2054.
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, Springer, London, (1999), 181-191. |
[21] |
Y. Tian, The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math., 25 (2002), 745-755.
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), 345-362. |
[23] |
W. Trench, Minimization problems for (R,S)-symmetric and (R,S)-skew symmetric matrices, Linear Algebra Appl., 389 (2004), 23-31.
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), 711-717.
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), 142-152.
doi: 10.1016/j.laa.2006.01.027. |
[27] |
L. Zadeh and C. Desoer, Linear System Theory: The State Space Approach, McGraw Hill, New York, 1963. |
[28] |
L. Zhang and D. Xie, A class of inverse eigenvalue problems, Math. Sci. Acta, 13 (1993), 94-99. |
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