**Matrix factorization** (or **matrix decomposition**) expresses a matrix as a product of simpler matrices, revealing structure and enabling efficient computation.

Common factorizations:

- **LU decomposition**: $A = LU$ (lower and upper triangular), used for solving linear systems
- **QR decomposition**: $A = QR$ (orthogonal and upper triangular), used for least squares
- **SVD**: $A = U\Sigma V^*$ (singular value decomposition), reveals rank and enables compression
- **Cholesky**: $A = LL^*$ (for positive-definite matrices)
- **Eigenvalue**: $A = PDP^{-1}$ (for diagonalizable matrices)
- **Schur**: $A = QTQ^*$ (unitary triangularization)
- **Rank factorization**: $A = CR$ where $C$ has full column rank and $R$ has full row rank

Each factorization has different computational costs, existence conditions, and applications.