Abstract
The interconnections and coupling patterns of dynamical systems are best described in terms of graph theory. This chapter serves the purpose of summarizing the main results and tools from matrix analysis and graph theory that will be important for the analysis of interconnected systems in subsequent chapters. This includes a proof of the Perron–Frobenius theorem for irreducible nonnegative matrices using a contraction mapping principle on convex cones due to Birkhoff (1957). We introduce adjacency matrices and Laplacians associated to a weighted directed graph and study their spectral properties. The analysis of eigenvalues and eigenvectors for graph adjacency matrices and Laplacians is the subject of spectral graph theory, which is briefly summarized in this chapter; see the book by Godsil and Royle (2001) for a comprehensive presentation. Explicit formulas for the eigenvalues and eigenvectors of Laplacians are derived for special types of graphs such as cycles and paths. These formulas will be used later on, in Chapter 9, in an examination of homogeneous networks. The technique of graph compression is briefly discussed owing to its relevance for the model reduction of networks. Properties of graphs are increasingly important for applications to, for example, formation control and molecular geometry. Therefore, a brief section is included on graph rigidity and the characterization of Euclidean distance matrices.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
B.D.O. Anderson, C. Yu, S. Dasgupta and A.S. Morse, “Control of a three-coleader formation in the plane”, Syst. Control Lett., vol. 56, (2007), 573–578.
L. Asimov and B. Roth, “The rigidity of graphs”, Trans. Am. Math. Soc., vol. 245, (1978), 279–289.
G. Birkhoff, “Extensions of Jentzsch’s Theorem”, Trans. Am. Math. Soc., vol. 85, (1957), 219–227.
L.M. Blumenthal, Theory and Applications of Distance Geometry, Cambridge University Press, Cambridge, UK, 1953.
S. Boyd, P. Diaconis and L. Xiao, “Fastest mixing Markov chain on a graph”, SIAM Review, vol. 46 (4), (2004), 667–689.
F. Bullo, J. Cortés and S. Martínez, Distributed Control of Robotic Networks, Princeton University Press, Applied Mathematics Series, 2009.
P.J. Bushell, “The Cayley–Hilbert metric and positive operators”, Linear Algebra Appl., vol. 84, (1986), 271–280.
R. Connelly, “Rigidity”, Ch. 1.7 of Handbook of Convex Geometry, Vol. A, (Ed. P. M. Gruber and J. M. Wills), Amsterdam, Netherlands: North-Holland, pp. 223–271, 1993.
G.M. Crippen and T.F. Havel, Distance Geometry and Molecular Conformation, J. Wiley and Sons, New York, 1988.
D. Cvetkovic, P. Rowlinson and S. Simic, An Introduction to the Theory of Graph Spectra, London Math. Soc. Student Texts 75, Cambridge University Press, Cambridge, UK, 2010.
P. J. Davis, Circulant Matrices, Wiley, New York, 1979.
F. Doerfler and F. Bullo, “Kron reduction of graphs with applications to electrical networks”, IEEE Trans. Circuits Syst. I, Reg. Papers, vol 60 (1), (2013), 150–163.
F. Doerfler and B. Francis, “Geometric analysis of the formation problem for autonomous robots”, IEEE Trans. Autom. Control, vol. 55 (2010), 2379–2384.
A.W.M. Dress and T.F. Havel, “Distance geometry and geometric algebra”, Foundations of Physics, vol. 23(10), (1993), 1357–1374.
M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics, Dover Publ. Inc., Mineola, NY, 2008.
W. Fulton, “Eigenvalues, invariant factors, highest weights, and Schubert calculus”, Bull. Am. Math. Soc., vol. 37, (2000), 209–249.
F.R. Gantmacher, The Theory of Matrices, 2 vols., Chelsea, 1959.
C.D. Godsil and G. Royle, Algebraic Graph Theory, Springer, New York, 2001.
J.C. Gower, “Properties of Euclidean and non-Euclidean distance matrices”, Linear Algebra Appl., vol. 67, (1985), 81–97.
U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, University of California Press, Berkeley and Los Angeles, 1958.
T.F. Havel and K. Wüthrich, “An evaluation of the combined use of nuclear magnetic resonance and distance geometry for the determination of protein conformations in solution”, J. Mol. Biol., vol. 182, (1985), 281–294.
U. Helmke and J.B. Moore, Optimization and Dynamical Systems, Springer-Verlag, London, 1994.
U. Helmke and J. Rosenthal, “Eigenvalue inequalities and the Schubert calculus”, Math. Nachrichten, vol. 171, (1995), 207–225.
L. Henneberg, Die graphische Statik der starren Systeme, B.G. Teubner, Leipzig, 1911.
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1990.
R.A. Horn, N.H. Rhee and W. So, “Eigenvalue inequalities and equalities”, Linear Algebra Appl., vol. 270, (1998), 29–44.
B. Huppert, Angewandte Linear Algebra, DeGruyter, Berlin, 1990.
R. Jentzsch, “Über Integraloperatoren mit positivem Kern”, Crelle J. Math., vol. 141, (1912), 235–244.
E. Kohlberg and J.W. Pratt, “The contraction mapping approach to the Perron–Frobenius theory: Why Hilbert’s metric?”, Mathematics for Operations Research, vol. 7 (2), (1982), 198–210.
I. Kra and S.R. Simanca, “On circulant matrices”, Notices of the Am. Math. Society, vol. 59, (2012), 368–377.
M.G. Krein and M.A. Rutman, “Linear operators leaving invariant a cone in a Banach space” Transl. Am. Math. Soc. Ser. 1, vol. 10, (1950), 199–325.
M.A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
G. Laman, “On graphs and the rigidity of plane skeletal structures”, J. Engineering Mathematics, vol. 4, (1970), 331–340.
M. Meshbahi and M. Egerstedt, Graph Theoretic Methods for Multiagent Networks, Princeton University Press, Princeton, NJ, 2010.
R. Olfati-Saber, J.A. Fax and R.M. Murray, “Consensus and cooperation in networked multi-agent systems”, Proc. IEEE, vol. 95, (2007), 215–233.
G. Ottaviani and B. Sturmfels, “Matrices with eigenvectors in a given subspace”, Proc. Am. Math. Soc. 141(4), (2013), 1219–1232,
B.N. Parlett and W.G. Poole, “A geometric theory for the QR, LU and power iterations”, SIAM J. Numer. Analysis, vol. 10, (1973), 389–412.
M. Pollicot and M. Yuri, Dynamical Systems and Ergodic Theory, London Math. Soc. Student Texts 40, Cambridge University press, Cambridge, U.K., 1998.
I.J. Schoenberg, “Remarks to Maurice Frechet’s article: Sur la definition axiomatique d’une classe d’espaces vectoriels distancies applicables vectoriellement sur l’espace de Hilbert” Ann. Math., vol. 36, (1935), 724–732.
E. Seneta, Non-negative matrices and Markov chains. 2nd rev. ed., XVI, Softcover Springer Series in Statistics, 1981.
S. Sternberg, Dynamical Systems, Dover Publications, Mineola, NY, 2010.
A.R. Willms, “Analytic results for the eigenvalues of certain tridiagonal matrices”, SIAM J. Matrix Anal. Appl., vol. 30, (2008), 639–656.
L. Xiao and S. Boyd, “Fast linear iterations for distributed averaging”, Syst. Control Lett., vol. 53, (2004), 65–78.
W.-C. Yueh, “Eigenvalues of several tridiagonal matrices”, Applied Mathematics E-Notes, vol. 5, (2005), 66–74.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Fuhrmann, P.A., Helmke, U. (2015). Nonnegative Matrices and Graph Theory. In: The Mathematics of Networks of Linear Systems. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-16646-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-16646-9_8
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16645-2
Online ISBN: 978-3-319-16646-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)