A note on positive de niteness and stability of interval matrices

Subject Areas : Applied Mathematics

H. Veiseh ¹

1 - Department of Applied Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran

Received: 2015-09-21 Accepted : 2015-09-21 Published : 2015-12-01

Keywords: Stability, Interval matrix, Real eigenvalues, Positive deniteness, Symmetric matrix,

Abstract :

It is proved that by using bounds of eigenvalues of an interval matrix, someconditions for checking positive deniteness and stability of interval matricescan be presented. These conditions have been proved previously with variousmethods and now we provide some new proofs for them with a unity method.Furthermore we introduce a new necessary and sucient condition for checkingstability of interval matrices.

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.