Handbook of Linear Algebra, 2nd Edition

Chapter 83

Some Applications of Matrices and Graphs in Euclidean Geometry

Miroslav Fiedler

Academy of Sciences of the Czech Republic

This chapter presents some facts and examples illustrating the interplay between matrix theory, graph theory, and n-dimensional Euclidean geometry. A recent global reference on the topic is the book [Fie11].

83.1 Euclidean Point Space

Definitions:

An arithmetic Euclidean vector space is the real inner product space ℝn with the standard inner product $\langle \mathbf{\text{x}},\mathbf{\text{y}}\rangle ={\mathbf{\text{y}}}^T\mathbf{\text{x}}$.

The arithmetic point Euclidean n-space En based on the vector space ℝn has as points the column n + 1-tuples with last coordinate 1, e.g., $\mathbf{\text{C}}={[c_1,c_2,...,c_n,1]}^T$, and as vectors the column n + 1-tuples with last coordinate 0, e.g., $\mathbf{\text{v}}={[{\upsilon }_1,{\upsilon }_2,...,{\upsilon }_n,0]}^T$.

The origin ...