Householder transformation

Summary

A Householder transformation is the reflection of $\R^n$ or $\C^n$ in a hyperplane through the origin. Its matrix is $I - 2 v v^*$ for a unit vector $v$ normal to the hyperplane. It is an orthogonal or unitary matrix, and in $\R^n$ it can send any given vector to any other vector of the same length. This makes it the basic tool for creating zeros in a matrix in a stable way. It is used in the QR decomposition and in the reduction to Hessenberg or tridiagonal form.