There have been several methods devoloped in recent years for decompositions of multisource data, consisting of more than one matrix with at least one shared dimension. Several methods have been developed for the simultaneous dimension reduction and decomposition of multiple matrices. Typically these methods assume that either features are shared for different sample sets (horizontal integration) or that samples are shared for different feature sets (vertical integration). However, these algorithms do not allow for simultaneous horizontal and vertical integration. For data sets that have shared sample sets and shared feature sets, we developed the Linked Matrix Factorization algorithm (LMF), an alternating least squares-based method that allows for decomposition of three matrices simultaneously when one matrix shares its sample set with one matrx and its feature set with another. We illustrate the application of LMF using a toxicology data set. In this data set, the toxicity matrix shares its sample set (cell lines) with a gene expression matrix, and it shares its feature set (chemicals) with a chemical attribute matrix.