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 Documentation / Mathematica / Built-in Functions / Lists and Matrices / Matrix Operations  /
QRDecomposition

  • QRDecomposition[ m ] yields the QR decomposition for a numerical matrix m. The result is a list q , r , where q is an orthogonal matrix and r is an upper triangular matrix.
  • The original matrix m is equal to Conjugate[Transpose[ q ]] . r.
  • For non-square matrices, q is row orthonormal.
  • The matrix r has zeros for all entries below the leading diagonal.
  • QRDecomposition[ m , Pivoting -> True] yields a list q , r , p where p is a permutation matrix such that m . p is equal to Conjugate[Transpose[ q ]] . r.
  • See the Mathematica book: Section 3.7.10.
  • See also Implementation NotesA.9.44.27MainBookLinkOldButtonDataA.9.44.27.
  • See also: SchurDecomposition, LUDecomposition, SingularValues, JordanDecomposition.
  • Related package: LinearAlgebra`Cholesky`, LinearAlgebra`Orthogonalization`.

    Further Examples

    Performing a QRDecomposition on this x matrix yields a pair of matrices.

    In[1]:=

    Out[1]=

    This checks the result.

    In[2]:=

    Out[2]//MatrixForm=