# zbMATH — the first resource for mathematics

The inverse eigenvalue problem for Hermitian anti-reflexive matrices and its approximation. (English) Zbl 1065.65057
A matrix $$A$$ is said to be anti-reflexive with respect to an orthogonal and symmetric matrix $$J$$ if $$A = -JAJ$$. Note that within the scope of inverse eigenvalue problems, it can be assumed without loss of generality that $J = \left[ \begin{matrix} I_r & 0 \\ 0 & -I_{n-r} \end{matrix} \right].$ In this case, $$A$$ is a Hermitian anti-reflexive matrix if and only if it takes the form $A = \left[ \begin{matrix} 0 & F \\ F^H & 0 \end{matrix} \right]$ for some $$r\times (n-r)$$ matrix $$F$$. This reveals a strong link between Hermitian anti-reflexive inverse eigenvalue problems and inverse singular value problems [cf. M. T. Chu, SIAM J. Numer. Anal. 29, No. 3, 885–903 (1992; Zbl 0757.65041)].
Given sets of vectors $$x_1,\dots,x_m$$ and scalars $$\lambda_1,\dots,\lambda_m$$, the author provides a complete characterization of $$S$$, the set of all Hermitian anti-reflexive matrices with eigenvector/eigenvalue pairs $$(x_i,\lambda_i)$$. Moreover, an explicit formula and a computational method for finding the matrix $$A^\star \in S$$ nearest to a given matrix $$A$$ are given. The paper is concluded by several numerical examples.

##### MSC:
 65F18 Numerical solutions to inverse eigenvalue problems
Full Text:
##### References:
  Baruch, M., Optimization procedure to correct stiffness and flexibility matrices using vibration tests, Aiaa j., 16, 1208-1210, (1978) · Zbl 0395.73056  Berman, A.; Nagy, E., Improvement of large analytical model using test data, Aiaa j., 21, 1168-1173, (1983)  Boley, D.; Golub, G.H., A survey of inverse eigenvalue problems, Inverse problem, 3, 595-622, (1987), (Printed in the UK) · Zbl 0633.65036  Chen, H.C., Generalized reflexive matrices: special properties and applications, SIAM J. matrix anal. appl., 19, 140-153, (1998) · Zbl 0910.15005  H.C. Chen, The SAS domain decomposition method for structural analysis, CSRD Tech. Report 754, Center for Super-computing Research and Development, University of Illinois, Urbana, IL, 1988  Chen, H.C.; Sameh, A., Numerical linear algebra algorithms on the ceder system, (), 101-125  Chen, J.L.; Chen, X.H., Special matrices (in Chinese), (2001), Qinghua University Press Beijing, China  Cheney, E., Introduction to approximation theory, (1966), McGraw-Hill · Zbl 0161.25202  Chu, M.T., Inverse eigenvalue problems, SIAM rev., 40, 1-39, (1998) · Zbl 0915.15008  Chu, M.T.; Erbrecht, M.A., Symmetric Toeplitz matrices with two prescribed eigenpairs, SIAM J. matrix anal. appl., 2, 623-635, (1994) · Zbl 0796.15025  Chu, M.T.; Golub, G., Structured inverse eigenvalue problems, Acta numer., 11, 1-71, (2002) · Zbl 1105.65326  Gladwell, G.M.L., The inverse problem for the vibrating beam, Proc. roy. soc., 393, 277-295, (1984) · Zbl 0542.73087  O. Hald, On Discrete and Numerical Sturm-Liouville Problems, Ph.D. Dissertation, New York University, New York, 1972  Higham, N.J., Computing a nearest symmetric positive semidefinite matrix, Linear algebra appl., 103, 103-118, (1988) · Zbl 0649.65026  Hochstad, H., On the construction of a Jacobi matrix from mixed given data, Linear algebra appl., 28, 113-115, (1979)  Jeseph, K.T., Inverse eigenvalue problem in structural design, Aiaa j., 30, 2890-2896, (1992) · Zbl 0825.73453  Jiang, Z.; Lu, Q., Optimal application of a matrix under spectral restriction, Math. numer. sinica, 1, 47-52, (1988)  Li, N.; Chu, K.W., Designing the Hopfield neural network via pole assignment, Internet, J. syst. sci., 25, 669-681, (1994) · Zbl 0805.93020  Meng, T., Experimental design and decision support, ()  The matrix equations AX=C, XB=D, Linear algebra appl., 59, 171-181, (1984) · Zbl 0543.15011  Sun, J.G., Two kinds of inverse eigenvalue problems for real symmetric matrices, Math. numer. sinica, 3, 282-290, (1988) · Zbl 0656.65041  Xie, D.X.; Zhang, L., Least-squares solution of inverse eigenvalue problems for anti-symmetric matrices, J. eng. math., 4, 25-34, (1993)  Xie, D.X.; Zhang, L.; Hu, X.Y., The solvability conditions for the inverse problem of bisymmetric nonnegative definite matrices, J. comput. math., 6, 597-608, (2000) · Zbl 0966.15008  Xu, S.F., An introduction to inverse algebraic eigenvalue problems, (1998), Peking University Press · Zbl 0927.65057  Zhou, F.Z.; Hu, X.Y.; Zhang, L., The solvability conditions for the inverse eigenvalue problems of centro-symmetric matrices, Linear algebra appl., 364, 147-160, (2003) · Zbl 1028.15012  Zhang, L., The approximation on the closed convex cone and its numerical application, Hunan ann. math., 6, 43-48, (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.