  We represent a two-qubit density matrix in the basis of Pauli matrix tensor
products, with the coefficients constituting a Bloch matrix, analogous to the
single qubit Bloch vector. We find the quantum state positivity requirements on
the Bloch matrix components, leading to three important inequalities, allowing
us to parametrize and visualize the two-qubit state space. Applying the
singular value decomposition naturally separates the degrees of freedom to
local and nonlocal, and simplifies the positivity inequalities. It also allows
us to geometrically represent a state as two entangled Bloch spheres with
superimposed correlation axes. It is shown that unitary transformations, local
or nonlocal, have simple interpretations as axis rotations or mixing of certain
degrees of freedom. The nonlocal unitary invariants of the state are then
derived in terms of local unitary invariants. The positive partial transpose
criterion for entanglement is generalized, and interpreted as a reflection, or
a change of a single sign. The formalism is used to characterize maximally
entangled states, and generalize two qubit isotropic and Werner states.
