A Householder matrix has eigenvalues . To see this, notice that if is orthogonal to the vector which was used to create the reflector, then , i.e., is an eigenvalue of multiplicity , since there are independent vectors orthogonal to . Also, notice , and so is an eigenvalue with multiplicity .
The determinant of a Householder reflector is , since the determinant of a matrix is the product of its eigenvalues, in this case one of which is with the remainder being (as in the previous point).
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 minors of that product.
This procedure is presented in Numerical Analysis by Burden and Faires. It uses a slightly altered function with .
In the first step, to form the Householder matrix in each step we need to determine and , which are:
From and , construct vector :
where , , and
Having found and computed the process is repeated for as follows:
Continuing in this manner, the tridiagonal and symmetric matrix is formed.
In this example, also from Burden and Faires, the given matrix is transformed to the similar tridiagonal matrix A3 by using the Householder method.
Following those steps in the Householder method, we have:
The first Householder matrix:
Used to form
As we can see, the final result is a tridiagonal symmetric matrix which is similar to the original one. The process is finished after two steps.
Computational and theoretical relationship to other unitary transformations
The Householder transformation is a reflection about a hyperplane with unit normal vector , as stated earlier. An -by-unitary transformation satisfies . Taking the determinant (-th power of the geometric mean) and trace (proportional to arithmetic mean) of a unitary matrix reveals that its eigenvalues have unit modulus. This can be seen directly and swiftly:
For the case of real valued unitary matrices we obtain orthogonal matrices, . It follows rather readily (see orthogonal matrix) that any orthogonal matrix can be decomposed into a product of 2 by 2 rotations, called Givens Rotations, and Householder reflections. This is appealing intuitively since multiplication of a vector by an orthogonal matrix preserves the length of that vector, and rotations and reflections exhaust the set of (real valued) geometric operations that render invariant a vector's length.
The Householder transformation was shown to have a one-to-one relationship with the canonical coset decomposition of unitary matrices defined in group theory, which can be used to parametrize unitary operators in a very efficient manner.
Finally we note that a single Householder transform, unlike a solitary Givens transform, can act on all columns of a matrix, and as such exhibits the lowest computational cost for QR decomposition and tridiagonalization. The penalty for this "computational optimality" is, of course, that Householder operations cannot be as deeply or efficiently parallelized. As such Householder is preferred for dense matrices on sequential machines, whilst Givens is preferred on sparse matrices, and/or parallel machines.