Breast cancer is heterogeneous and consists of intrinsic components with various alterations. Combining multiply genomic sources to identify the intrinsic components and their heterogeneity is essential for precise clinical decision-making. In this study, we explore the heterogeneous intrinsic hallmarks of breast cancer by decomposing the integrated tensor of gene expression, copy number alteration, and DNA methylation information. This is achieved using advanced Bayesian tensor decomposition, which can extract multi-level latent representatives for both patients and genes. Gene set enrichment analysis is applied to estimate the key biological functions of the identified intrinsic genomic components. Patient stratification and survival analysis are performed based on the heterogeneity of these functional genomic components. The findings are evaluated using The Cancer Genome Atlas Breast Carcinoma (TCGA-BRCA) dataset. This study could provide insight into how multi-omics can identify functional intrinsic components of cancer and stratify patients into prognostically significant groups.