**Volume 17, Issue 4, pp 477-487**

**Publication Date**: 2017-10-14

http://dx.doi.org/10.22436/jmcs.017.04.04

Li-Jun Zhao - School of Mathematics and Information Engineering, Taizhou University, Linhai, Zhejiang, 317000, P. R. China

Ru Huang - Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322, USA

This article considers an inverse eigenvalue problem for centrosymmetric matrices under a central principal submatrix constraint and the corresponding optimal approximation problem. We first discuss the specified structure of centrosymmetric matrices and their central principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the inverse eigenvalue problem, and we derive an expression for its general solution. Finally, we obtain an expression for the solution to the corresponding optimal approximation problem.

Centrosymmetric matrix, central principal submatrix, inverse eigenvalue problem, optimal approximation problem

[1] A. L. Andrew, Eigenvectors of certain matrices, Linear Algebra and Appl., 7 (1973), 157–162.

[2] A. Borobia, R. Canogar, The real nonnegative inverse eigenvalue problem is NP-hard, Linear Algebra Appl., 522 (2017), 127–139.

[3] M. T. Chu, Inverse eigenvalue problems, SIAM Rev., 40 (1998), 1–39.

[4] L.-F. Dai, M.-L. Liang, W.-Y. Ma, Optimization problems on the rank of the solution to left and right inverse eigenvalue problem, J. Ind. Manag. Optim., 11 (2015), 171–183.

[5] L. Datta, S. D. Morgera, On the reducibility of centrosymmetric matrices–applications in engineering problems, Circuits Systems Signal Process., 8 (1989), 71–96.

[6] P. Deift, T. Nanda, On the determination of a tridiagonal matrix from its spectrum and a submatrix, Linear Algebra Appl., 60 (1984), 43–55.

[7] G. H. Golub, C. F. Van Loan, Matrix computations, Third edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, (1996).

[8] I. J. Good, The inverse of a centrosymmetric matrix, Technometrics, 12 (1970), 925–928.

[9] X. Y. Hu, L. Zhang, Least-square approximation solutions to a class of matrix problems, (Chinese) Hunan Daxue Xuebao, 17 (1990), 98–102.

[10] Z. X. Jiang, Q. S. Lu, On optimal approximation of a matrix under a spectral restriction, (Chinese) Math. Numer. Sinica, 8 (1986), 47–52.

[11] K. T. Joseph, Inverse eigenvalue problem in structural design, AIAA J., 30 (1992), 2890–2896.

[12] M. Kimura, Some problems of stochastic processes in genetics, Ann. Math. Statist., 28 (1957), 882–901.

[13] N. Li, A matrix inverse eigenvalue problem and its application, Linear Algebra Appl., 266 (1997), 143–152.

[14] Z.-Y. Peng, X.-Y. Hu, L. Zhang, The inverse problem of centrosymmetric matrices with a submatrix constraint, J. Comput. Math., 22 (2004), 535–544.

[15] Y. M. Ram, J. Caldwell, Physical parameters reconstruction of a free-free mass-spring system from its spectra, SIAM J. Appl. Math., 52 (1992), 140–152.

[16] J. R. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors, Amer. Math. Monthly, 92 (1985), 711–717.

[17] J. Xu, D. X. Xie, Central symmetric least squares solution for matrix equation with a submatrix constraint, J. B. Inf. Sci. Technol. Univ., 31 (2016), 31–35.

[18] Q.-X. Yin, Construction of real antisymmetric and bi-antisymmetric matrices with prescribed spectrum data, Linear Algebra Appl., 389 (2004), 95–106.

[19] L. A. Zhornitskaya, V. S. Serov, Inverse eigenvalue problems for a singular Sturm-Liouville operator on [0, 1], Inverse Problems, 10 (1994), 975–987.

[20] F.-Z. Zhou, X.-Y. Hu, L. Zhang, The solvability conditions for the inverse eigenvalue problems of centro-symmetric matrices, Linear Algebra Appl., 364 (2003), 147–160.