  Here we introduce the non-Hermitian diluted banded random matrix (nHdBRM)
ensemble as the set of $N\times N$ real non-symmetric matrices whose entries
are independent Gaussian random variables with zero mean and variance one if
$|i-j|<b$ and zero otherwise, moreover off-diagonal matrix elements within the
bandwidth $b$ are randomly set to zero such that the sparsity $\alpha$ is
defined as the fraction of the $N(b-1)/2$ independent non-vanishing
off-diagonal matrix elements. By means of a detailed numerical study we
demonstrate that the eigenfunction and spectral properties of the nHdBRM
ensemble scale with the parameter $x=\gamma[(b\alpha)^2/N]^\delta$, where
$\gamma,\delta\sim 1$. Moreover, the normalized localization length $\beta$ of
the eigenfunctions follows a simple scaling law: $\beta = x/(1 + x)$. For
comparison purposes, we also report eigenfunction and spectral properties of
the Hermitian diluted banded random matrix ensemble.
