Applications
In geometric optics, specular reflection can be expressed in terms of the Householder matrix.
Householder reflections can be used to calculate QR decompositions by reflecting first one column of a matrix onto a multiple of a standard basis vector, calculating the transformation matrix, multiplying it with the original matrix and then recursing down the (i, i) minors of that product.
They are also widely used for tridiagonalization of symmetric matrices and for transforming non-symmetric matrices to a Hessenberg form.
Read more about this topic: Householder Transformation