  We consider the Bloch-Torrey operator, $-\Delta + igx$, that governs the time
evolution of the transverse magnetization in diffusion magnetic resonance
imaging (dMRI). Using the matrix formalism, we compute numerically the
eigenvalues and eigenfunctions of this non-Hermitian operator for two bounded
three-dimensional domains: a sphere and a capped cylinder. We study the
dependence of its eigenvalues and eigenfunctions on the parameter $g$ and on
the shape of the domain (its eventual symmetries and anisotropy). In
particular, we show how an eigenfunction drastically changes its shape when the
associated eigenvalue crosses a branch (or exceptional) point in the spectrum.
Potential implications of this behavior for dMRI are discussed.
