Left and Right Inverse Eigenpairs Problem of Orthogonal Matrices

Affiliation(s)

Institute of Mathematics and Physics, School of Sciences, Central South University of Forestry and Technology,.

Institute of Mathematics and Physics, School of Sciences, Central South University of Forestry and Technology,.

ABSTRACT

In this paper, the left and right inverse eigenpairs problem of orthogonal matrices and its optimal approximation solution are considered. Based on the special properties of eigenvalue and the special relations of left and right eigenpairs for orthogonal matrices, we find the equivalent problem, and derive the necessary and sufficient conditions for the solvability of the problem and its general solutions. With the properties of continuous function in bounded closed set, the optimal approximate solution is obtained. In addition, an algorithm to obtain the optimal approximation and numerical example are provided.

In this paper, the left and right inverse eigenpairs problem of orthogonal matrices and its optimal approximation solution are considered. Based on the special properties of eigenvalue and the special relations of left and right eigenpairs for orthogonal matrices, we find the equivalent problem, and derive the necessary and sufficient conditions for the solvability of the problem and its general solutions. With the properties of continuous function in bounded closed set, the optimal approximate solution is obtained. In addition, an algorithm to obtain the optimal approximation and numerical example are provided.

Cite this paper

F. Li, "Left and Right Inverse Eigenpairs Problem of Orthogonal Matrices,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 1972-1976. doi: 10.4236/am.2012.312271.

F. Li, "Left and Right Inverse Eigenpairs Problem of Orthogonal Matrices,"

References

[1] J. H. Wilkinson, “The Algebraic Eigenvalue Problem,” Oxford University Press, Oxford, 1965.

[2] M. Arav, D. Hershkowitz, V. Mehrmann, et al., “The Recursive Inverse Eigenvalue Problem,” SIAM Journal on Matrix Analysis and Applications, Vol. 22, No. 2, 2000, pp. 392-412.

[3] R. Loewy and V. Mehrmann, “A Note on the Symmetric Recursive Inverse Eigenvalue Problem,” SIAM Journal on Matrix Analysis and Applications, Vol. 25, No. 1, 2003, pp. 180-187.

[4] L. Zhang and D. X. Xie, “A Class of Inverse Eigenvalue Problems,” Acta Mathematica Scientia, Vol. 1, No. 13, 1993, pp. 94-99.

[5] F. L. Li, X. Y. Hu and L. Zhang, “Left and Right Eigenpairs Problem of Skew-Centrosymmetric Matrices,” Applied Mathematics and Computation, Vol. 177, No. 1, 2006, pp. 105-110.

[6] F. L. Li, X. Y. Hu and L. Zhang, “Left and Right Inverse Eigenpairs Problem of Generalized Centrosymmetric Matrices and Its Optimal Approximation Problem,” Applied Mathematics and Computation, Vol. 212, No. 1, 2009, pp. 481-487. doi:10.1016/j.amc.2009.02.035

[7] F. L. Li and K. K. Zhang, “Left and Right Inverse Eigenpairs Problem for the Symmetrizable Matrices,” Proceedings of the Ninth International Conference on Matrix Theory and Its Applications, Vol. 1, 2010, pp. 179-182.

[8] M. L. Liang and L. F. Dai, “The Left and Right Inverse Eigenvalue Problems of Generalized Reflexive and Anti-Reflexive Matrices,” Journal of Computational and Applied Mathematics, Vol. 234, No. 3, 2010, pp. 743-749.

[9] H. Dai, “The Theory of Matrices,” Science Press, Beijing, 2001.

[1] J. H. Wilkinson, “The Algebraic Eigenvalue Problem,” Oxford University Press, Oxford, 1965.

[2] M. Arav, D. Hershkowitz, V. Mehrmann, et al., “The Recursive Inverse Eigenvalue Problem,” SIAM Journal on Matrix Analysis and Applications, Vol. 22, No. 2, 2000, pp. 392-412.

[3] R. Loewy and V. Mehrmann, “A Note on the Symmetric Recursive Inverse Eigenvalue Problem,” SIAM Journal on Matrix Analysis and Applications, Vol. 25, No. 1, 2003, pp. 180-187.

[4] L. Zhang and D. X. Xie, “A Class of Inverse Eigenvalue Problems,” Acta Mathematica Scientia, Vol. 1, No. 13, 1993, pp. 94-99.

[5] F. L. Li, X. Y. Hu and L. Zhang, “Left and Right Eigenpairs Problem of Skew-Centrosymmetric Matrices,” Applied Mathematics and Computation, Vol. 177, No. 1, 2006, pp. 105-110.

[6] F. L. Li, X. Y. Hu and L. Zhang, “Left and Right Inverse Eigenpairs Problem of Generalized Centrosymmetric Matrices and Its Optimal Approximation Problem,” Applied Mathematics and Computation, Vol. 212, No. 1, 2009, pp. 481-487. doi:10.1016/j.amc.2009.02.035

[7] F. L. Li and K. K. Zhang, “Left and Right Inverse Eigenpairs Problem for the Symmetrizable Matrices,” Proceedings of the Ninth International Conference on Matrix Theory and Its Applications, Vol. 1, 2010, pp. 179-182.

[8] M. L. Liang and L. F. Dai, “The Left and Right Inverse Eigenvalue Problems of Generalized Reflexive and Anti-Reflexive Matrices,” Journal of Computational and Applied Mathematics, Vol. 234, No. 3, 2010, pp. 743-749.

[9] H. Dai, “The Theory of Matrices,” Science Press, Beijing, 2001.