It is proved that by using bounds of eigenvalues of an interval matrix, some conditions for checking positive de niteness and stability of interval matrices can be presented. These conditions have been proved previously with various methods and now we provide some new proofs for them with a unity method. Furthermore we introduce a new necessary and sucient condition for checking stability of interval matrices.
Article Title [Persian]
A note on positive de niteness and
stability of interval matrices
Authors [Persian]
Hana Veiseh
Department of Applied Mathematics, Hamedan Branch, Islamic Azad
University, Hamedan, Iran
Abstract [Persian]
It is proved that by using bounds of eigenvalues of an interval matrix, some conditions for checking positive de niteness and stability of interval matrices can be presented. These conditions have been proved previously with various methods and now we provide some new proofs for them with a unity method. Furthermore we introduce a new necessary and sucient condition for checking stability of interval matrices.
Keywords [Persian]
Interval matrix، Real eigenvalues، Positive de niteness، Stability، Symmetric matrix
References
[1] J. Rohn, Checking positive de niteness or stability of symmetric interval matrices is NP-hard, Commentationes Mathematicae Universitatis Carolinae. 35 (1994) 795{797. [2] J. Rohn, Checking properties of interval matrices, Technical Report 686, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, September 1996. [3] R. Farhadsefat, T. Lot , J. Rohn, A note on regularity and positive de niteness of interval matrices, Cent. Eur. J. Math. 10(1) (2012) 322{328. [4] M. Mansour, Robust stability of interval matrices, Proceeding of the 28th Conference on Decision and Control, Tampa, FL. (1989) 46{51. [5] J. Rohn, A Handbook of Results on Interval Linear Problems, Prague: Czech Academy of Sciences, 2005. [6] J. Rohn, Positive de niteness and stability of interval matrices, SIAM J. Matrix Anal. Appl. 15(1) (1994) 175{184. [7] S. Poljak, J. Rohn, Checking roboust nonsingularity is NP-hard, Math. Control Signals Syst. 6(1) (1993) 1{9. [8] A. S. Deif, The interval eigenvalue problem, Z. Angew. Math. Mech. 71(1) (1991) 61{64. [9] M. Hladik, D. Daney, E. P. Tsigaridas, Bounds on real eigenvalues and singular values of interval matrices, SIAM J. Matrix Anal. Appl. 31(4) (2010) 2116{2129.
[10] J. Rohn, Bounds on eigenvalues of interval matrices, ZAMM, Z. Angew. Math. Mech. 78(Suppl. 3) (1998) S1049{S1050. [11] J. Rohn, Interval matrices: Singularity and real eigenvalues, SIAM Journal on Matrix Analysis and Applications. 14(1) (1993) 82{91. [12] J. Rohn, A. Deif, On the range of eigenvalues of an interval matrix, Comput. 47(3-4) (1992) 373{377. [13] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge: Cambridge University Press, 1985. [14] J. Stoer and R. Bulrisch, Introduction to Numerical Analysis, Springer-Verlag, Berlin, 1980.
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