# Calculus: Compute a QR Factorization using the Householder Method

**Compute a QR Factorization of A using the Householder Method. Separately list all of the Householder reflections used along the way.**

A =

Start by finding the Householder reflection M with V1 = the first column of A
X = V1 = (1, 2, 2)
Y = ||X||e1 = (3, 0, 0)
U = (Y-X)/(||Y-X||) = (1/sqrt(3))(1, -1, -1)
M1 = I – 2 UU^t (^t denotes the matrix’s transpose)
M1 =
1/3 | 2/3 | 2/3 |

2/3 | 1/3 | -2/3 |

2/3 | -2/3 | 1/3 |

**Solution: A = QR**
**R = M1A = **
**Q = M1^t =**
1/3 | 2/3 | 2/3 |

2/3 | 1/3 | -2/3 |

2/3 | -2/3 | 1/3 |