Abstract
Pairs of linear transformations on a finite dimensional vector space are of great relevance in the analysis of two-dimensional (2D) systems evolutions. In this paper, special properties of matrix pairs, such as finite memory, separability, property L and property P, as well as their dynamical interpretations, are investigated. Practical criteria for testing property L and property P in a finite number of steps are also presented.
The nonnegativity requirement on a matrix pair allows for much stronger characterizations of finite memory and separability, which in fact prove to be structural properties. Finally, the irreducibility and primitivity notions of positive matrix pairs are discussed, and connected with the dynamical behavior of the associated 2D systems.
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References
A. Berman and R.J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, New York (NY ), (1979).
M. Bisiacco, E. Fornasini, and G. Marchesini, 2D partial fraction expansions and minimal commutative realizations, IEEE Trans. Circ. Sys. 35 (1988), pp. 1533–1538.
M. Fliess, Un outil algébrique: Les séries formelles non commutatives, Springer Lect.Notes in Econ. Math. Sys. 131, (1975), pp. 122–149.
E. Fornasini and G. Marchesini, Doubly indexed dynamical systems, Math. Sys. Theory 12, (1978), pp. 59–72.
E. Fornasini and G. Marchesini, Properties of pairs of matrices and state-models for 2D systems, In C.R.Rao, editor, Multivariate Analysis: Future Directions. North Holland Series in Probability and Statistics 5, (1993), pages 131–80.
E. Fornasini, G. Marchesini, and M.E. Valcher, On the structure of finite memory and separable 2D systems, Automatica 29, (1994), pp. 347–356.
E. Fornasini and M.E. Valcher, Matrix pairs in 2D systems: an approach based on trace series and Hankel matrices, SIAM J. Contr. Optim. 33, 4 (1995), pp. 1127–1150.
E. Fornasini and M.E. Valcher, On the spectral and combinatorial structure of 2D positive systems, Lin. Alg. Appl. 245, (1996), pp. 223–258.
E. Fornasini and M.E. Valcher, Directed graphs, 2D state models and characteristic polynomial of irreducible matrix pairs, to appear in Lin. Alg. Appl., (1997).
E. Fornasini and M.E. Valcher, Recent developments in 2D positive system theory, to appear in J. of Applied Mathematics, (1997).
W.P. Heath. Self-tuning control for two-dimensional processes, J.Wiley and Sons, (1994).
N.H. McCoy, On the characteristic roots of matrix polynomials, Bull. Amer.Math.Soc. 42, (1936), pp. 592–600.
H. Minc, Nonnegative Matrices, J.Wiley and Sons, New York, (1988).
T.S. Motzkin and O. Taussky, Pairs of matrices with property L, Trans. Amer. Math.Soc. 73, (1952), pp. 108–114.
T.S. Motzkin and O. Taussky, Pairs of matrices with property L (II), Trans. Amer. Math.Soc. 80, (1955), pp. 387–401.
C. Procesi, The invariant theory of n x n matrices. Advances in Methematics 19, (1976), pp. 306–381.
A. Salomaa and M. Soittola, Automata theoretic aspects of formal power series, Springer-Verlag, (1978).
D. Shemesh, Common eigenvectors of two matrices, Linear Algebra Appl. 62, (1984), pp. 11–18.
S. Vomiero, Un’applicazione dei sistemi 2D alla modellistica dello scambio sangue-tessuto, PhD thesis, (in Italian), Univ.di Padova, Italy, (1992).
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© 1997 Springer Fachmedien Wiesbaden
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Fornasini, E., Marchesini, G., Valcher, M.E. (1997). Matrix Pairs and 2D Systems Analysis. In: Helmke, U., Prätzel-Wolters, D., Zerz, E. (eds) Operators, Systems and Linear Algebra. European Consortium for Mathematics in Industry. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-09823-2_6
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DOI: https://doi.org/10.1007/978-3-663-09823-2_6
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
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