Householder Transformation - Computational and Theoretical Relationship To Other Unitary Transformations

Computational and Theoretical Relationship To Other Unitary Transformations

The Householder Transformation is a reflection about a certain hyperplane, namely, the one with unit normal vector v, as stated earlier. An N by N unitary transformation U satisfies UUH=I. Taking determinant (N-th power of the geometric mean) and trace (proportional to arithmetic mean) of a unitary matrix reveals that its eigenvalues λ_i are unit modulus. This can be seen directly and swiftly:

Since arithmetic and geometric means are equal iff the variables are constant, see, inequality of arithmetic and geometric means, we establish the claim of unit modulus.

For the case of real valued unitary matrixes we obtain orthogonal matrices, In this case all eigenvalues are real, and so the unit modulus eigenvalue constraint is replaced by the binary constraint that all eigenvalues lie in the set {+1,-1}. 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.