The objective of this dissertation was to evaluate, by simulation and by livestock datasets (Holstein, Jersey, Angus, swine, and broiler chicken), the dimensionality of genomic information in closed populations, and the effect of using regular or APY (Algorithm for Proven and Young) inverses of the genomic relationship matrices (GRM) on genomic predictions. Additionally, APY in combination with pedigree truncation was tested to improve convergence in a livestock dataset experiencing slow convergence. A GRM can be inverted efficiently with APY through recursion on a small number of core animals, with the number of core animals theoretically linked to effective population size (Ne). Eigenvalue decomposition of the GRM was used to determine the numbers of largest eigenvalues corresponding to 90, 95, 98, and 99% of variation in the GRM. The rank of the GRM was equal to or less than the number of genotyped animals and the number of single nucleotide polymorphism (SNP) markers in the study. Data were analyzed using the APY inverse of the GRM with randomly selected core animals, and the number of core animals was equal to the number of largest eigenvalues. Realized accuracies peaked with the number of core animals corresponding to 98 to 99% of the variation depending on population, indicating that 1 to 2% of the variation in the GRM is due to noise. Assuming the optimum number of core animals is equal to the dimensionality of the genomic information, that dimensionality was about 14,000 for Holstein and Angus cattle, 12,000 for Jersey cattle, and 6000 for swine and broiler chickens, which corresponds approximately to 3NeL. With a genome length of 30 Morgans, approximate Ne was 149 for Holsteins, 101 for Jerseys, 113 for Angus, and 44 for broiler chickens; with a genome length of 20 Morgans, approximate Ne was 48 for swine. Limited dimensionality caused improved convergence when the APY inverse was used, as this inverse used the information in GRM more efficiently than the regular inverse.