Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30076
The Inverse Problem of Nonsymmetric Matrices with a Submatrix Constraint and its Approximation

Authors: Yongxin Yuan, Hao Liu


In this paper, we first give the representation of the general solution of the following least-squares problem (LSP): Given matrices X ∈ Rn×p, B ∈ Rp×p and A0 ∈ Rr×r, find a matrix A ∈ Rn×n such that XT AX − B = min, s. t. A([1, r]) = A0, where A([1, r]) is the r×r leading principal submatrix of the matrix A. We then consider a best approximation problem: given an n × n matrix A˜ with A˜([1, r]) = A0, find Aˆ ∈ SE such that A˜ − Aˆ = minA∈SE A˜ − A, where SE is the solution set of LSP. We show that the best approximation solution Aˆ is unique and derive an explicit formula for it. Keyw

Keywords: Inverse problem, Least-squares solution, model updating, Singular value decomposition (SVD), Optimal approximation.

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1285


[1] J. P. Aubin, Applied Functional Analysis, John Wiley, New York, 1979.
[2] Z. J. Bai, The inverse eigenproblem of centrosymmetric matrices with a submatrix constraint and its approximation, SIAM J. Matrix Anal. Appl. 26 (2005) 1100-1114.
[3] K. J. Bathe, E. L. Wilson, Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1976.
[4] A. Ben-Israel, T. N. E. Greville, Generalized Inverse: Theory and Applications, John Wiley, New York, 1974.
[5] A. Berman, Mass matrix correction using an incomplete set of measured modes, AIAA Journal. 17 (1979) 1147-1148.
[6] P. D. Cha, W. Gu, Model updating using an incomplete set of experimental modes, Journal of Sound and Vibration. 233 (2000) 587-600.
[7] M. T. Chu, G. H. Golub, Inverse Eigenvalue Problems, Theory, Algorithms and Applications, Oxford Uinversity Press, 2005.
[8] R. G. Cobb, B. Liebst, Structural damage identification using assigned partial eigenstructure, AIAA Journal. 35 (1997) 152-158.
[9] H. Dai, About an inverse eigenvalue problem arising in vibration analysis, RAIRO Math. Model. Numer. Anal. 29 (1995) 421-434.
[10] H. Dai, P. Lancaster, Linear matrix equations from an inverse problem of vibration theory, Linear Algebra Appl. 246 (1996) 31-47.
[11] M. I. Friswell, J. E. Mottershead, Finite Element Moodel Updating in Structural Dynamics, Klumer Academic Publishers, Dordrecht, 1995.
[12] H. Liu and H. Dai, Inverse problems for nonsymmetric matrices with a submatrix constraint, Appl. Math. Comput. 189 (2007) 1320-1330.
[13] J. C. O-Callahan, C. M. Chou, Localization of model errors in optimized mass and stiffness matrices using modal test data, Internat. J. Analytical and Experimental Analysis. 4 (1989) 8-14.
[14] Z. Y. Peng, X. Y. Hu and L. Zhang, The inverse problem of bisymmetric matrices with a submatrix constraint, Numer. Linear Algebra Appl. 11 (2003) 59-73.
[15] F. S. Wei, Stiffness matrix correction from incomplete test data, AIAA Journal. 18 (1980) 1274-1275.
[16] F. S. Wei, Analytical dynamic model improvement using vibration test data, AIAA Journal. 28 (1990) 174-176.
[17] H. Q. Xie, Sensitivity analysis of eigenvalue problems, Ph. D. thesis, Nanjing University of Aeronautics and Astronautics, 2003.
[18] S. F. Xu, An Introduction to Inverse Algebraic Eiegnvalue Problems, Peking University Press, Beijing, 1998.
[19] Y. X. Yuan, H. Dai, Inverse problems for symmetric matrices with a submatrix constraint, Appl. Numer. Math. 57 (2007) 646-656.
[20] Q. Q. Zhang, A. Zerva, D. W. Zhang, Stiffness matrix adjustment using incomplete measured modes, AIAA Journal. 35 (1997) 917-919.