Matrices and Graphs in Geometry

by Miroslav Fiedler

• ISBN13: 9780521461931
• ISBN10: 0521461936
• Format: Hardcover
• Copyright: 2011-03-14
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Preface	p. vii
A matricial approach to Euclidean geometry	p. 1
Euclidean point space	p. 1
n-simplex	p. 4
Some properties of the angles in a simplex	p. 12
Matrices assigned to a simplex	p. 15
Simplex geometry	p. 24
Geometric interpretations	p. 24
Distinguished objects of a simplex	p. 31
Qualitative properties of the angles in a simplex	p. 46
Signed graph of a simplex	p. 46
Signed graphs of the faces of a simplex	p. 50
Hyperacute simplexes	p. 52
Position of the circumcenter of a simplex	p. 53
Special simplexes	p. 64
Right simplexes	p. 64
Orthocentric simplexes	p. 72
Cyclic simplexes	p. 94
Simplexes with a principal point	p. 108
The regular n-simplex	p. 112
Further geometric objects	p. 114
Inverse simplex	p. 114
Simplicial cones	p. 119
Regular simplicial cones	p. 131
Spherical simplexes	p. 132
Finite sets of points	p. 135
Degenerate simplexes	p. 142
Applications	p. 145
An application to graph theory	p. 145
Simplex of a graph	p. 145
Geometric inequalities	p. 147
Extended graphs of tetrahedrons	p. 152
Resistive electrical networks	p. 153
Appendix	p. 159
Matrices	p. 159
Graphs and matrices	p. 175
Nonnegative matrices, M- and P-matrices	p. 179
Hankel matrices	p. 182
Projective geometry	p. 182
Reference	p. 193
Index	p. 195
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