GRAM–SCHMIDT ORTHONORMAL TRANSFORMATION CLASSICAL GRAM–SCHMIDT ORTHONORMAL TRANSFORMATION The Gram–Schmidt orthonormalization procedure was introduced in Section in order to introduce the orthonormal transformation F applied to the matrix T. The Gram–Schmidt orthonormalization procedure described there is called the classical Gram–Schmidt (CGS) orthogonilization procedure. The CGS procedure was developed in detail for the case s ¼ 3, m 0 ¼ 2, and then these results were extrapolated to the general case of arbitrary s and m 0 . In this section we shall develop the CGS procedure in greater detail. The CGS procedure is sensitive to computer round-off errors. There is. | 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 13 GRAM-SCHMIDT ORTHONORMAL TRANSFORMATION CLASSICAL GRAM-SCHMIDT ORTHONORMAL TRANSFORMATION The Gram-Schmidt orthonormalization procedure was introduced in Section in order to introduce the orthonormal transformation F applied to the matrix T. The Gram-Schmidt orthonormalization procedure described there is called the classical Gram-Schmidt CGS orthogonilization procedure. The CGS procedure was developed in detail for the case s 3 m 2 and then these results were extrapolated to the general case of arbitrary s and m . In this section we shall develop the CGS procedure in greater detail. The CGS procedure is sensitive to computer round-off errors. There is a modified version of the Gram-Schmidt procedure that is not sensitive to computer round-off errors. This is referred to as the modified Gram-Schmidt MGS . After developing the general CGS results we shall develop the MGS procedure. One might ask why explain the CGS procedure at all. It is better to start with a description of the CGS procedure because first it is simpler to explain second it makes it easier to obtain a physical feel for the Gram-Schmidt orthogonalization and third it provides a physical feel for its relationship to the Householder and in turn Givens transformations. Hence we shall first again start with a description of the CGS procedure. As described in Section starting with the m 1 vectors 11 12 . tm 1 we transform these to m 1 orthogonal vectors which we designate as q 1 q 2 . q m 1. Having this orthogonal set the desired orthonormal set of Section see Figure can be obtained by dividing qi by its magnitude qi to form qi. We start by picking the first vector q 1 equal to 11 322 CLASSICAL GRAM-SCHMIDT ORTHONORMAL TRANSFORMATION 323 that is q 1 11 At this point the matrix T o 1 1 12 tm i can be thought of as being transformed to the
