Four equivalent sets of mixed-model equations with relationship matrix for estimation of genetic parameters

Four equivalent approaches of estimating variance components from mixed-model equations including the relationship matrix are presented. One procedure involves simultaneous diagonalization of the coefficient matrix (Z'MZ) and the inverse of the numerator relationship matrix (A-1). After simultaneous diagonalization, the absorbed coefficient matrix of the mixed-model equations is diagonal, so solving the mixed model equations and estimating variance components become trivial computations. A numerical example is given to illustrate the computational aspects of the simultaneous diagonalization approach and to demonstrate that the four approaches yield identical results. The simultaneous diagonalization approach eliminates direct inversion of the coefficient matrix for each iteration and thus requires much less computer time. The advantage of this approach over that of direct matrix inversion in terms of computer time increases with increasing levels of random effects and increasing rounds of iteration. Application of simultaneous diagonalization to maximum likelihood estimation also is discussed.