Adaptive Filters

CHAPTER 34 Unitary Transformations

In this chapter we describe two classes of unitary transformations (Givens and Householder) that can be used to annihilate selected entries in a vector or matrix and thereby reduce a matrix to triangular (or similar) form, as is often required by array algorithms. Special care needs to be taken when dealing with complex-valued data as compared to real-valued data.

34.1 GIVENS ROTATIONS

Givens rotations provide an effective way to annihilate specific entries in a vector and it is enough to explain their operation on 2-dimensional row vectors. We consider the case of real-valued data first.

Real data

Consider a 1 × 2 real-valued vector z = [ a b ], and assume that we wish to determine a 2 × 2 matrix Θ that transforms it to the form:

(34.1)

for some real number α to be determined, and where Θ is required to be orthogonal, i.e., it should satisfy

We refer to [a b] as the pre-array and to [α 0] as the post-array.

Now any orthogonal matrix Θ has the important property that it preserves vector norms. Indeed, it is easy to see from (34.1) that the following equality must hold:

or, equivalently, a² + b² = α². In this way, no matter which orthogonal ...