  We study the linear stability of weakly magnetized differentially rotating
plasmas in both collisionless kinetic theory and Braginskii's theory of
collisional, magnetized plasmas. We focus on the very weakly magnetized limit
that is important for understanding how astrophysical magnetic fields originate
and are amplified at high redshift. We show that the single instability of
fluid theory - the magnetorotational instability mediated by magnetic tension -
is replaced by two distinct instabilities, one associated with ions and one
with electrons. Each of these has a different way of tapping into the free
energy of differential rotation. The ion instability is driven by viscous
transport of momentum across magnetic field lines due to a finite ion cyclotron
frequency (gyroviscosity); the fastest growing modes have wavelengths
significantly longer than MHD and Hall MHD predictions. The electron
instability is a whistler mode driven unstable by the temperature anisotropy
generated by differential rotation; the growth time can be orders of magnitude
shorter than the rotation period. The electron instability is an example of a
broader class of instabilities that tap into the free energy of differential
rotation or shear via the temperature anisotropy they generate. We briefly
discuss the application of our results to the stability of planar shear flows
and show that such flows are linearly overstable in the presence of fluid
gyroviscosity. We also briefly describe the implications of our results for
magnetic field amplification in the virialized halos of high redshift galaxies.
