In this paper we introduce a new method for analyzing expression patterns from high throughput and complex data such as gene expression microarrays. These microarrays are collected under different conditions such as time, phenotype and treatment. The proposed method uses a Bayesian matrix decomposition, called Bayesian linear unmixing (BLU), to extract a set of characteristic gene signatures, or factors, and a set of coefficients, factor scores, that specify the relative contribution of each signature to a specific sample. BLU is related to Bayesian factor analysis but differs in an important respect: BLU constrains the factor loadings to be non-negative and the factor scores to be probability distributions over the factors. Thus BLU reduces the multiplexing of genes into different factors and can enhance interpretability of the factor loadings and factor scores. The unsupervised version of BLU presented in this paper also provides estimates of the number of factors. We illustrate the application of BLU to bioinformatics by analyzing gene expression microarray datasets.