The single-step covariance matrix H combines the pedigree-based relationship matrix with the more accurate information on realized relatedness of genotyped individuals represented by the genomic relationship matrix . In particular, to improve convergence behavior of iterative approaches and to reduce inflation, two weights and have been introduced in the definition of , which blend the inverse of a part of with the inverse of . Since the definition of this blending is based on the equation describing , its impact on the structure of is not obvious. In a joint discussion, we considered the question of the shape of for non-trivial and .

Here, we present the general matrix as a function of these parameters and discuss its structure and properties. Moreover, we screen for optimal values of and with respect to predictive ability, inflation and iterations up to convergence on a well investigated, publicly available wheat data set.

Our results may help the reader to develop a better understanding for the effects of changes of and on the covariance model. In particular, we give theoretical arguments that as a general tendency, inflation will be reduced by increasing or by decreasing .