The Jacquard genetic identity coefficients are of fundamental importance in relatedness research. We address the estimation of these coefficients as well as other relationship parameters that derive from them such as kinship and inbreeding coefficients using a concise matrix framework. Estimation of the Jacquard coefficients via likelihood methods and the expectation-maximization algorithm is computationally very demanding for large numbers of polymorphisms. We propose a constrained least squares approach to estimate the Jacquard coefficients. A simulation study shows constrained least squares achieves root-mean-squared errors that are comparable with those of the maximum likelihood approach, in particular when founder allele frequencies are unknown, while obtaining enormous computational savings.