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Inverse of an Additive x Additive Relationship Matrix Due to Sires and Maternal Grandsires

Department of Animal Sciences, University of Illinois, Urbana 61801

ABSTRACT

A method for inverting an additive x additive relationship matrix due to sire and maternal grandsires is presented. It is based on a recursive relationship between these epistatic effects in offspring and male ancestors (sires and maternal grandsires) expressed in terms of a transition matrix between generations. The required inverse is a function of this matrix and of the inverse of the variance-covariance matrix of segregation residuals (R). This matrix is at least as sparse as the additive x additive relationship matrix. Without inbreeding, R is often diagonal or block diagonal, so the calculations required are easier than those needed for direct inversion.