  A finite non-commutative geometry consists of a fuzzy space together with a
Dirac operator satisfying the axioms of a real spectral triple. This paper
addreses the question of how to extract information about these geometries from
the spectrum of the Dirac operator. Since the Dirac operator is a
finite-dimensional matrix, the usual asymptotics of the eigenvalues makes no
sense and is replaced by measurements of the spectrum at a finite energy scale.
The spectral dimension of the square of the Dirac operator is improved to
provide a new spectral measure of the dimension of a space called the spectral
variance. Similarly, the volume of a space can be computed from the spectrum
once the dimension is known. Two methods of doing this are investigated: the
well-known Dixmier trace and a recent improvement due to Abel Stern. Finally,
the distance between two geometries is investigated by comparing the spectral
zeta functions using the method of Cornelissen and Kontogeorgis. All of these
techniques are tested on the explicit examples of the fuzzy spheres and fuzzy
tori, which can be regarded as approximations of the usual Riemannian sphere
and flat tori. Then they are applied to characterise some random fuzzy spaces
using data generated by a Monte Carlo simulation.
