  Using the first-principles density functional approach, we investigate a
material Pr$_2$MgIrO$_6$ (PMIO) of double perovskite structure synthesized
recently. According to the calculations, PMIO is a magnetic Mott-Hubbard
insulator influenced by the cooperative effect of spin-orbit coupling (SOC) and
Coulomb interactions of Ir-5$d$ and Pr-4$f$ electrons, as well as the crystal
field. When Pr is replaced with Sr gradually, the system exhibits half metallic
(HM) states desirable for spintronics applications. In
[Pr$_{2-x}$Sr$_x$MgIrO$_6$]$_2$, HM antiferromagnetism (HMAFM) with zero spin
magnetization in the unit cell is obtained for $x=1$, whereas for $x=0.5$ and
1.5 HM ferrimagnetism (HMFiM) is observed with $\mu_{\rm tot}=3\mu_{\rm B}$ and
$\mu_{\rm tot}=-3\mu_{\rm B}$ per unit cell respectively. It is emphasized that
the large exchange splitting between spin-up and spin-down bands at the Fermi
level makes the half metallicity possible even with strong SOC.
