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.
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- Definition of singular value decomposition (SVD)Every $A\in\mathbb C^{m\times n}$ has a singular value decomposition $$A=U\Sigma V^ ,$$ where $U,V$ are square...
- LU decompositionDefinition: The LU decomposition of a square matrix $A$ is a factorization: $$A = LU$$ where $L$ is a lower...
- Definition of rank factorization of a matrixDefinition: A rank factorization of an $m \times n$ matrix $A$ of rank $r$ is a decomposition: $$A = CR$$ where $C$...
- Definition of Cholesky decompositionDefinition: The Cholesky decomposition of a symmetric (Hermitian) positive-definite matrix $A$ is a factorization:...
- QR decompositionDefinition: The QR decomposition of an $m \times n$ matrix $A$ (with $m \geq n$) is a factorization: $$A = QR$$...
- Definition of Schur triangulationDefinition: Schur triangulation (or Schur decomposition ) states that every square matrix $A$ over $\mathbb{C}$ can...
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