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Structural Properties of Polynomial and Rational Matrices, a survey

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Abstract (2. Language): 
A review of the structural properties of polynomial and rational ma- trices is presented. After the analysis of the finite spectrum of a polyno- mial matrix A(), via the Smith canonical form, we analyze the infinity as an eigenvalue, but also as a pole or zero (via the Smith McMillan canonical form) when considering A() in the set of rational matrices. Then we focus on the structures generated by the columns of A(). Here we review two different approaches: when considering linear com- binations over the rational functions, and when linear combinations are supposed to be over polynomials only. The objective is to compare and contrast the results of these two lines of thought, as well as to under- line the fundamental differences between matrix polynomials in one or several variables. Structure preserving transformations and equivalence relations over polynomial matrices are also reviewed. With this objec- tive in mind, we also give some insights on the eigenvalue structure of multivariable matrix polynomials.
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REFERENCES

References: 

[1] P.J. Antsaklis and Z. Gao, Polynomial and rational matrix interpolation:
theory and control applications, Int. J. Control, vol. 58 pp. 349–404,
1993.
[2] G.D. Birkhoff and S. MacLane, A Survey of Modern Algebra Third Edition,
Macmillan, New York, 1965.
[3] D. Cox, J. Little and D. O’Shea, Ideals, Varieties and Algorithms: an
introduction to computational Algebraic Geometry and Commutative
Algebra, Springer, New York, 1997.
[4] D. Cox, J. Little and D. O’Shea, Using Algebraic Geometry, Graduate
Texts in Mathematics 185, Springer, New York, 1998.
[5] G.D. Forney, Minimal bases of rational vector spaces, with applications
to multivariable linear systems, SIAM Journal on Control, vol. 13, pp.
493–520, 1975.
[6] F.R. Gantmacher, Theory of Matrices, Chelsea, New York, 1959.
[7] I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic
Press, New York, 1982.
[8] I. Gohberg, P. Lancaster, and L. Rodman, Invariant Subspaces of Matrices
with Applications, Wiley, New York, 1986 and SIAM, Philadelphia,
2006.
402 Juan C. Z´u˜niga–Anaya
[9] B. Hartley and T.O. Hawkes, Rings, Modules and Linear Algebra, Chapman
and Hall, London, 1970.
[10] T. Kailath, Linear Systems, Prentice Hall, Englewood Cliffs, 1980.
[11] N. Karampetakis and S. Vologiannidis, Infinite elementary divisor structure
preserving transformations for polynomial matrices, Int. J. Appl.
Math. Comput. Sci. vol. 13, pp. 493–503, 2003.
[12] V. Kuˇcera, Discrete Linear Control: The Polynomial Equation Approach,
John Wiley and Sons, Chichester, 1979.
[13] V. Kuˇcera, Diophantine equations in control–a survey, Automatica, vol.
29, pp. 1361–1375, 1993.
[14] P. Lancaster, Linearization of regular matrix polynomials, Electronic
Journal of Linear Algebra, vol. 17, pp. 21–27, 2008.
[15] J. J. Loiseau, Sur la modification de la structure `a l’infini par un retour
d’´etat statique, SIAM J. Control Optim., vol. 26, pp. 251–273, 1988.
[16] C.C. MacDuffee, The Theory of Matrices, Chelsea, New York, 1952.
[17] B. McMillan, Introduction to formal realizability theory, Bell System
Tech. J., vol. 31, pp. 217–219, 541–600, 1952.
[18] A. S. Morse, Structural invariants of linear multivariable systems, SIAM
J. Control Optim., vol. 11, pp. 446–465, 1973.
[19] W. H. L. Neven and C. Praagman, Column reduction of polynomial
matrices, Linear Algebra Appl., vol. 188, pp. 569–589, 1993.
[20] J. W. Polderman and J. C. Willems, Introduction to Mathematical Systems
Theory: a Behavioral Approach, Springer-Verlag, 1998.
[21] H.H. Rosenbrock, State-space and Multivariable Theory, John Wiley,
New York, 1970.
[22] H.J.S. Smith, On systems of linear indeterminate equations and congruences,
Philosophical Transactions of the Royal Society, vol. 151, pp.
293–326, 1861.
[23] H.J. Stetter, Numerical Polynomial Algebra, SIAM, Philadelphia, 2004.
[24] P. M. Van Dooren, The computation of Kronecker’s canonical form of a
singular pencil, Linear Algebra Appl. vol. 27, pp. 103–140, 1979.
Structural Properties of Polynomial and Rational Matrices, a survey 403
[25] P. M. Van Dooren and P. Dewilde, The Eigenstructure of an Arbitrary
Polynomial Matrix. Computational Aspects, Linear Algebra Appl., vol.
50, pp. 545–580, 1983.
[26] A.I.G. Vardulakis, Linear Multivariable Control, JohnWiley, Chichester,
UK, 1991.
[27] W.A. Wolovich, Linear Multivariable Systems, Springer Verlag, 1974.
[28] W. M. Wonham and A. S. Morse, Decoupling and pole assignment in
linear multivariable systems: A geometric approach. SIAM J. Control
Optim. vol. 8, pp. 1–18, 1970.
[29] J.C. Z´u˜niga Anaya and D. Henrion, An improved Toeplitz algorithm
for polynomial matrix null-space computation, Appl. Mathematics and
Comp., vol. 207, pp. 256–272, 2009.

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