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  1. Home
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  3. Matrices and Graphs in Geometry
  • Cited by 11
Publisher:
Cambridge University Press
Online publication date:
June 2013
Print publication year:
2011
Online ISBN:
9780511973611
DOI:
https://doi.org/10.1017/CBO9780511973611
Subjects:
Mathematics (general), Mathematics, Discrete Mathematics Information Theory and Coding, Geometry and Topology
Series:
Encyclopedia of Mathematics and its Applications (139)
Information

Book description

Simplex geometry is a topic generalizing geometry of the triangle and tetrahedron. The appropriate tool for its study is matrix theory, but applications usually involve solving huge systems of linear equations or eigenvalue problems, and geometry can help in visualizing the behaviour of the problem. In many cases, solving such systems may depend more on the distribution of non-zero coefficients than on their values, so graph theory is also useful. The author has discovered a method that in many (symmetric) cases helps to split huge systems into smaller parts. Many readers will welcome this book, from undergraduates to specialists in mathematics, as well as non-specialists who only use mathematics occasionally, and anyone who enjoys geometric theorems. It acquaints the reader with basic matrix theory, graph theory and elementary Euclidean geometry so that they too can appreciate the underlying connections between these various areas of mathematics and computer science.

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Contents

Contents
References
References
References
[1] L.M., Blumenthal: Theory and Applications of Distance Geometry. Oxford, Clarendon Press, 1953.
[2] E., Egerváry: On orthocentric simplexes. Acta Math. Szeged IX (1940), 218–226.
[3] M., Fiedler: Geometrie simplexu I. Časopis pěst. mat. 79 (1954), 270–297.
[4] M., Fiedler: Geometrie simplexu II. Časopis pěst. mat. 80 (1955), 462–476.
[5] M., Fiedler: Geometrie simplexu III. Časopis pěst. mat. 81 (1956), 182–223.
[6] M., Fiedler: Über qualitative Winkeleigenschaften der Simplexe. Czechosl. Math. J. 7(82) (1957), 463–478.
[7] M., Fiedler: Einige Sätze aus der metrischen Geometrie der Simplexe in Euklidischen Räumen. In: Schriftenreihe d. Inst. f. Math. DAW, Heft 1, Berlin (1957), 157.
[8] M., Fiedler: A note on positive definite matrices. (Czech, English summary.)Czechosl. Math. J. 10(85) (1960), 75–77.
[9] M., Fiedler: Über eine Ungleichung für positive definite Matrizen. Mathematische Nachrichten 23 (1961), 197–199.
[10] M., Fiedler: Über die qualitative Lage des Mittelpunktes der umgeschriebenen Hyperkugel im n-Simplex. Comm. Math. Univ. Carol. 2(1) (1961), 3–51.
[11] M., Fiedler: Über zyklische n-Simplexe und konjugierte Raumvielecke. Comm. Math. Univ. Carol. 2(2) (1961), 3–26.
[12] M., Fiedler, V., Pták: On matrices with non-positive off-diagonal elements and positive principal minors. Czechosl. Math. J. 12(87) (1962), 382–400.
[13] M., Fiedler: Hankel matrices and 2-apolarity. Notices AMS 11 (1964), 367–368.
[14] M., Fiedler: Relations between the diagonal elements of two mutually inverse positive definite matrices. Czechosl. Math. J. 14(89) (1964), 39–51.
[15] M., Fiedler: Some applications of the theory of graphs in the matrix theory and geometry. In: Theory of Graphs and Its Applications. Proc. Symp. Smolenice 1963, Academia, Praha (1964), 37–41.
[16] M., Fiedler: Matrix inequalities. Numer. Math. 9 (1966), 109–119.
[17] M., Fiedler: Algebraic connectivity of graphs. Czechosl. Math. J. 23(98) (1973), 298–305.
[18] M., Fiedler: Eigenvectors of acyclic matrices. Czechosl. Math. J. 25(100) (1975), 607–618.
[19] M., Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechosl. Math. J. 25(100) (1975), 619–633.
[20] M., Fiedler: Aggregation in graphs. In: Coll. Math. Soc. J. Bolyai, 18. Combinatorics. Keszthely (1976), 315–330.
[21] M., Fiedler: Laplacian of graphs and algebraic connectivity. In: Combinatorics and Graph Theory, Banach Center Publ. vol. 25, PWN, Warszava (1989), 57–70.
[22] M., Fiedler: A geometric approach to the Laplacian matrix of a graph. In: Combinatorial and Graph-Theoretical Problems in Linear Algebra (R. A., Brualdi, S., Friedland, V., Klee, editors), Springer, New York (1993), 73–98.
[23] M., Fiedler: Structure ranks of matrices. Linear Algebra Appl. 179 (1993), 119–128.
[24] M., Fiedler: Elliptic matrices with zero diagonal. Linear Algebra Appl. 197, 198 (1994), 337–347.
[25] M., Fiedler: Moore–Penrose involutions in the classes of Laplacians and simplices. Linear Multilin. Algebra 39 (1995), 171–178.
[26] M., Fiedler: Some characterizations of symmetric inverse M-matrices. Linear Algebra Appl. 275–276 (1998), 179–187.
[27] M., Fiedler: Moore-Penrose biorthogonal systems in Euclidean spaces. Linear Algebra Appl. 362 (2003), 137–143.
[28] M., Fiedler: Special Matrices and Their Applications in Numerical Mathematics, 2nd edn, Dover Publ., Mineola, NY (2008).
[29] M., Fiedler, T. L., Markham: Rank-preserving diagonal completions of a matrix. Linear Algebra Appl. 85 (1987), 49–56.
[30] M., Fiedler, T. L., Markham: A characterization of the Moore–Penrose inverse. Linear Algebra Appl. 179 (1993), 129–134.
[31] R. A., Horn, C. A., Johnson: Matrix Analysis, Cambridge University Press, New York, NY (1985).
[32] D. J. H., Moore: A geometric theory for electrical networks. Ph.D. Thesis, Monash. Univ., Australia (1968).

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