Tracking and Kalman filtering made easy P11

GIVENS ORTHONORMAL TRANSFORMATION THE TRANSFORMATION The Givens orthonormal transformation for making a matrix upper triangular is made up of successive elementary Givens orthonormal transformations G 1 ; G 2 ; . . . to be defined shortly. Consider the matrix T expressed by 2 3 t 11 t 12 t 13 6t t t 7 ð11:1-1Þ T ¼ 6 21 22 23 7 4 t 31 t 32 t 33 5 t 41 t 42 t 43 First, using the simple Givens orthonormal transformation matrix G 1 the matrix T is transformed to 2 3 ðt 11 Þ 1 ðt. | 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 11 GIVENS ORTHONORMAL TRANSFORMATION THE TRANSFORMATION The Givens orthonormal transformation for making a matrix upper triangular is made up of successive elementary Givens orthonormal transformations G 1 G 2 . to be defined shortly. Consider the matrix T expressed by 111 t12 t13 121 t22 t23 131 t32 t33 141 t42 t43 11-1-1 First using the simple Givens orthonormal transformation matrix G1 the matrix T is transformed to r t 11 1 t12 1 t13 1 G1T 0 t22 1 t23 1 131 132 133 141 142 143 The transformation G1 forces the 2 1 term of the matrix T to be zero. Now applying another elementary Givens orthonormal transformation G2 to the above matrix yields G 2G1T t11 2 t12 2 t13 2 0 t22 1 t23 1 60 t32 2 t33 2 141 142 143 283 284 GIVENS ORTHONORMAL TRANSFORMATION The second transformation G2 forces the 3 1 term to be zero. Applying in turn the third Givens orthonormal transformation G 3 to the above matrix now yields G 3G 2G iT t11 3 t12 3 t13 3 0 t22 1 t 23 1 0 t32 2 t33 2 0 t42 3 t 43 3_ 11-1-4 Application of these successive elementary Givens orthonormal transformations has forced all the elements of the first column of T to zero below the first element. This process is now repeated for the second column of the above matrix with another set of elementary Givens orthonormal transformations so as to force all the elements below the diagonal of the second column to be zero. This process is next repeated for the third and last column of the matrix so as to force the elements below the diagonal to be zero yielding the desired upper triangular matrix expressed by 2 U M 11 u 12 M13 0 u 22 u 23 H H 0 0 0 M 33 0 0 0 The first elementary Givens orthonormal transformation G 1 above is given as G1 c 1 -s 1 s 1 1 c 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0 1 where c 1 cos d 1 t11 t 21 121 1 2 and S 1 sin d 1 t21 .