A key problem in biomedical research is to elucidate the complex gene regulatory network underlying complex traits such as common human diseases.
 In genetical genomics (eQTL) experiments, gene expression levels are often treated as quantitative traits that are subject to genetic analysis. 
 These data can also provide important information on gene regulation and genetic networks.
 In this talk, I introduce a sparse high dimensional multivariate regression model for studying the conditional independent relationships
 among a set of genes adjusting for possible genetic effects, as well as the genetic architecture that influences the gene expression. 
 I present a covariate adjusted precision matrix estimation method (CAPME), which can be easily implemented by linear programming.  
Asymptotic convergence rates and sign consistency are established for estimators of the regression coefficients and the precision matrix. 
 Numerical performance of the estimator is investigated using both simulated and real data sets.  
Simulation results have shown that the CAPME results in great improvements in both estimation and graph structure selection.
   We apply CAPME to analysis of a yeast eQTL data in order to identify the gene regulatory network among a set of genes in the MAPK pathway. 
 In addition, I will discuss analysis of multi-tissue eQTL data and simultaneous estimation of multiple precision matrices with similar structures.