Large sparse matrices—such as adjacency matrices of web or social graphs, feature matrices in machine‑learning pipelines, and graph‑database relations—are routinely compressed to save space. Classical compressors require full decompression before use, turning space savings into computational overhead. Recent work studies computation‑friendly compression: formats that remain small while allowing matrix‑vector products, and in some cases matrix‑matrix products, sums and closures, to be performed directly on the compressed representation in time proportional to its size.\n\nThe literature can be organized by the redundancy each format exploits:\n- Spatial clustering of non‑zeros using $k^2$‑trees and variants;\n- Row similarity via Boldi‑Vigna‑style referencing or set‑difference compression;\n- Repeated substrings captured by grammar compression;\n- Dense bipartite substructures represented as bicliques.\n\nFor each family we describe the representation, algorithms, theoretical guarantees and empirical performance. Moreover, the last three families are instances of a single scheme called sparse straight‑line factorization (SSLF). With SSLF, forward and backward substitution evaluate right and left products in $O(\text{representation size})$ time; the product of two SSLFs is again an SSLF, yielding matrix‑matrix multiplication with a compressed output. The model connects to linear circuits, admits simple recompression steps, and allows study of the smallest possible factorization—providing worst‑case bounds, separations from each family, and parallelism considerations.\n\nThe survey concludes with a cross‑family comparison and a list of open problems.\n\nReview: This work offers a coherent overview of computation‑friendly matrix compression, unifying disparate techniques under a common factorization framework and pointing to promising directions for scalable graph and learning applications.