  We constrain the luminosity and redshift dependence of the intrinsic
alignment (IA) of a nearly volume-limited sample of luminous red galaxies
selected from the fourth public data release of the Kilo-Degree Survey
(KiDS-1000). To measure the shapes of the galaxies, we used two complementary
algorithms, finding consistent IA measurements for the overlapping galaxy
sample. The global significance of IA detection across our two independent
luminous red galaxy samples, with our favoured method of shape estimation, is
$\sim10.7\sigma$. We find no significant dependence with redshift of the IA
signal in the range $0.2<z<0.8$, nor a dependence with luminosity below
$L_r\lesssim 2.9 \times 10^{10} h^{-2} L_{r,\odot}$. Above this luminosity,
however, we find that the IA signal increases as a power law, although our
results are also compatible with linear growth within the current
uncertainties. This behaviour motivates the use of a broken power law model
when accounting for the luminosity dependence of IA contamination in cosmic
shear studies.
