HOUSEHOLDER ORTHONORMAL TRANSFORMATION In the preceding chapter we showed how the elementary Givens orthonormal transformation triangularized a matrix by successfully zeroing out one element at a time below the diagonal of each column. With the Householder orthonormal transformation all the elements below the diagonal of a given column are zeroed out simultaneously with one Householder transformation. Specifically, with the first Householder transformation H 1 , all the elements below the first element in the first column are simultaneously zeroed out, resulting in the form given by (). With the second Householder transformation H 2 all the elements below the second element. | Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright 1998 John Wiley Sons Inc. ISBNs 0-471-18407-1 Hardback 0-471-22419-7 Electronic 12 HOUSEHOLDER ORTHONORMAL TRANSFORMATION In the preceding chapter we showed how the elementary Givens orthonormal transformation triangularized a matrix by successfully zeroing out one element at a time below the diagonal of each column. With the Householder orthonormal transformation all the elements below the diagonal of a given column are zeroed out simultaneously with one Householder transformation. Specifically with the first Householder transformation H1 all the elements below the first element in the first column are simultaneously zeroed out resulting in the form given by . With the second Householder transformation H2 all the elements below the second element of the second column are zeroed out and so on. The Householder orthonormal transformation requires fewer multiplies and adds than does the Givens transformation in order to obtain the transformed upper triangular matrix of 103 . The Householder transformation however does not lend itself to a systolic array parallel-processor type of implementation as did the Givens transformation. Hence the Householder may be preferred when a centralized processor is to be used but not if a custom systolic signal processor is used. COMPARISON OF HOUSEHOLDER AND GIVENS TRANSFORMATIONS Let us initially physically interpret the Householder orthonormal transformation as transforming the augmented matrix T0 to a new coordinate system as we did for the Givens transformations. For the first orthonormal Householder transformation H1 the s rows are unit vectors designated as hi for the ith row onto 315 316 HOUSEHOLDER ORTHONORMAL TRANSFORMATION which the columns of T0 are projected. The first-row vector h1 is chosen to line up with the vector formed by the first column of To designated as 11. Hence the projection of 11 onto this coordinate yields a value for the .
