Based on the nonlinear Perron-Frobenius theory
, we will develop the theory to rigorously formalize the model from a mathematical viewpoint (existence, uniqueness, maximality).
Samelson developed independently in considering Perron-Frobenius Theory
in finite dimensions.
By the Perron-Frobenius theory
, [rho](G) has multiplicity one and exists a unique positive unit eigenvector corresponding to [rho](G).
Sections start with coverage of vector spaces, progressing to linear operators and matrices, the duality of vector spaces, determinants, invariant subspaces, inner-product spaces, structure theorems, and additional topics such as functions of an operator, quadratic forms, Perron-Frobenius theory
, stochastic matrices, and representations of finite groups.
In addition to theoretical foundation and the latest work in Hilbert geometry, this overview for students and researchers covers relationships between Hilbert geometry and other subjects in mathematics, such as convexity theory, Perron-Frobenius theory
, partial differential equations, ergodic theory, and Lie groups.
By the well-known Perron-Frobenius theory
, [rho](G) is simple and has a unique positive unit eigenvector and so does q(G).