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By Erik W Grafarend; Joseph L Awange

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Extra resources for Applications of linear and nonlinear models : fixed effects, random effects, and total least squares

Example text

48) Proof. 50) constitute the necessary conditions. 51), according to the positive semi-definiteness of the matrix Gx , generate the sufficiency condition for obtaining the minimum of the constrained Lagrangean. A/ the existence of Gx -MINOS xm is guaranteed. In order to prove uniqueness of Gx -MINOS xm , we have to consider case (i) (Gx positive-definite) and we have to consider case (ii) (Gx positive semidefinite). 51) Case (i) (Gx positive-definite): Due to rkGx D m and jGx j ¤ 0, the partitioned system of normal equations (ref) is uniquely solved.

115), namely the rank partitioning of the canonical unknown vector x into x1 2 Rr and x2 2 Rm r to determine x1 D ƒ 1 y , but leaving x2 underdetermined. Consult the commutative diagram that is shown in Fig. 7 for a short hand summary of the introduced transformations of coordinates of both the parameter space X and the observation space Y. Next we shall proof that x2 D 0 if x is MINOS. 8, our basic result on canonical MINOS, subsequently completed by proofs. 7. (eigenspace analysis versus eigenspace synthesis of the matrix fA 2 Rn m ; r WD rkA D n < mg) The pair of matrices fL; Rg for the eigenspace analysis and the eigenspace synthesis of the rectangular matrix A 2 Rn m of rank r WD rkA D n < m, namely 1-2 Minimum Norm Solution (MINOS) Fig.

121) 2 r Proof. Let us prove first AA# L D Lƒ2 , second A# AR1 D R1 ƒ2 . 8. (The canonical MINOS). Let y D A x be a canonical representation of the underdetermined system of linear equations fy D AxjA 2 Rn m ; r WD rkA D n; n < mg. Then the rank partitioning of xm is Gx -MINOS. 127). 128). 128) Proof. 58) and replace the matrix A 2 m by its canonical representation, namely eigenspace synthesis. 133) x2 0 0 0 The pair of eigensystems AA# L D Lƒ2 ; A# AR1 D R1 ƒ2 is unfortunately based upon non-symmetric matrices AA# D AGx 1 A0 Gy and A# A D Gx 1 A0 Gy A which make the left and right eigenspace analysis numerically more complex.

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