Advances in molecular "omics'" technologies have motivated new methodology for the integration of multiple sources of high-content biomedical data. However, most statistical methods for integrating multiple data matrices only consider data shared vertically (one cohort on multiple platforms) or horizontally (different cohorts on a single platform). This is limiting for data that take the form of bidimensionally linked matrices (e.g., multiple cohorts measured on multiple platforms), which are increasingly common in large-scale biomedical studies. In this paper, we propose BIDIFAC (Bidimensional Integrative Factorization) for integrative dimension reduction and signal approximation of bidimensionally linked data matrices. Our method factorizes the data into (i) globally shared, (ii) row-shared, (iii) column-shared, and (iv) single-matrix structural components, facilitating the investigation of shared and unique patterns of variability. For estimation we use a penalized objective function that extends the nuclear norm penalization for a single matrix. As an alternative to the complicated rank selection problem, we use results from random matrix theory to choose tuning parameters. We apply our method to integrate two genomics platforms (mRNA and miRNA expression) across two sample cohorts (tumor samples and normal tissue samples) using the breast cancer data from TCGA. We provide R code for fitting BIDIFAC, imputing missing values, and generating simulated data.