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Factorization of matrices

Created over 8 years ago, updated 10 days ago

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.