  We develop an in-depth analysis of the $SO(4)$ Landau models on $S^3$ in the
$SU(2)$ monopole background and their associated matrix geometry. The Schwinger
and Dirac gauges for the $SU(2)$ monopole are introduced to provide a concrete
coordinate representation of $SO(4)$ operators and wavefunctions. The gauge
fixing enables us to demonstrate algebraic relations of the operators and the
$SO(4)$ covariance of the eigenfunctions. With the spin connection of $S^3$, we
construct an $SO(4)$ invariant Weyl-Landau operator and analyze its eigenvalue
problem with explicit form of the eigenstates. The obtained results include the
known formulae of the free Weyl operator eigenstates in the free field limit.
Other eigenvalue problems of variant relativistic Landau models, such as
massive Dirac-Landau and supersymmetric Landau models, are investigated too.
With the developed $SO(4)$ technologies, we derive the three-dimensional matrix
geometry in the Landau models. By applying the level projection method to the
Landau models, we identify the matrix elements of the $S^3$ coordinates as the
fuzzy three-sphere. For the non-relativistic model, it is shown that the fuzzy
three-sphere geometry emerges in each of the Landau levels and only in the
degenerate lowest energy sub-bands. We also point out that Dirac-Landau
operator accommodates two fuzzy three-spheres in each Landau level and the mass
term induces interaction between them.
