QRDecomposition[m] yields the QR decomposition for a numerical matrix m. The result is a list {q, r}, where q is a unitary matrix and r is an upper-triangular matrix.

The decomposition of a 2×2 matrix into a unitary (orthogonal) matrix q and upper triangular matrix r: In[1]:=m=(1 | 2 3 | 4); {q, r}=QRDecomposition[m]; {q//MatrixForm, r//MatrixForm} Out[1]={(1/sqrt(10) | 3/sqrt(10) 3/sqrt(10) | -1/sqrt(10)), (sqrt(10) | 7 sqrt(2/5) 0 | sqrt(2/5))} Verify that m = q^†.r: In[2]:=m==ConjugateTranspose[q].r Out[2]=True Compute the QR decomposition for a 3×2 matrix with exact values: In[1]:={q, r}=QRDecomposition[(1 | 2 3 | 4 5 | 6)]; {q//MatrixForm, r//MatrixForm} Out[1]={(1/sqrt(35) | 3/sqrt(35) | sqrt(5/7) 13/sqrt(210) | 2 sqrt(2/105) | -sqrt(5/42)), (sqrt(35) | 44/sqrt(35) 0 | 2 sqrt(6/35))} q^T.r is the original matrix: In[2]:=Transpose[q].r//MatrixForm Out[2]=(1 | 2 3 | 4 5 | 6) Compute the QR decomposition for a 2×3 matrix with approximate numerical values: In[1]:={q, r}=QRDecomposition[(1. | 2. | 3. 4. | 5. | 6.)]; {q//MatrixForm, r//MatrixForm} Out[1]={(-0.242536 | -0.970143 -0.970143 | 0.242536), (-4.12311 | -5.33578 | -6.54846 0. | -0.727607 | -1.45521)} q^T.r is the original matrix: In[2]:=Transpose[q].r//MatrixForm Out[2]=(1. | 2. | 3. 4. | 5. | 6.)

Pivoting | TargetStructure

SchurDecomposition | LUDecomposition | SingularValueDecomposition | JordanDecomposition | CholeskyDecomposition | HessenbergDecomposition | Orthogonalize

introduced in Version 2 (January 1991) last modified in Version 14 (December 2023)