# Matrix Factorization

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## Abstract

Just as multiplication can be generalized from scalars to matrices, the notion of factorization can also be generalized from scalars to matrices. Exact matrix factorizations need to satisfy the size and rank constraints that are imposed on matrix multiplication. For example, when an *n* × *d* matrix *A* is factorized into two matrices *B* and *C* (i.e., *A* = *BC*), the matrices *B* and *C* must be of sizes *n* × *k* and *k* × *d* for some constant *k*. For exact factorization to occur, the value of *k* must be equal to at least the rank of *A*. This is because the rank of *A* is at most equal to the minimum of the ranks of *B* and *C*. In practice, it is common to perform *approximate* factorization with much smaller values of *k* than the rank of *A*.

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