Orthogonalisation Procedures

Abstract

In Chapter 8, we have seen that Gauss (1809, §§ 181-183) had related the partial derivatives of the adjusted quadratic forms,½Ω, ½Ω₁, ½Ω₂ etc and

$$\begin{array}{*{20}{r}} {{\xi _0} = }&{{u_{00}}x + {u_{01}}y + {u_{02}}z + etc + {s_0}} \\ {{\eta _1} = }&{{u_{11}}y + {u_{12}}z + etc + {s_1}} \\ {{\zeta _2} = }&{{u_{22}}z + etc + {s_2}} \\ {etc}&{} \end{array}$$

to the linear combinations of the errors

$$\begin{array}{*{20}{l}} {\begin{array}{*{20}{c}} {\xi = }&{\sum {{a_i}{v_i}} } \end{array}} \\ {\begin{array}{*{20}{c}} {\eta = }&{\sum {{b_i}{v_i}} } \end{array}} \\ {\begin{array}{*{20}{c}} {\zeta = }&{\sum {{c_i}{v_i}} } \end{array}} \\ {etc} \end{array}$$

by means of the inverse relationships

$$\begin{array}{*{20}{l}} {\begin{array}{*{20}{c}} {\xi = }&{{\xi _0}} \end{array}} \\ {\begin{array}{*{20}{c}} {\eta = }&{\frac{{{u_{01}}}}{{{u_{00}}}}{\xi _0} + {\eta _1}} \end{array}} \\ {\begin{array}{*{20}{c}} {\zeta = }&{\frac{{{u_{02}}}}{{{u_{00}}}}{\xi _0} + \frac{{{u_{12}}}}{{{u_{11}}}}{\eta _1} + {\zeta _2}} \end{array}} \\ {etc} \end{array}$$

and thus directly to each other by means of the equations

$$\begin{array}{*{20}{l}} {\begin{array}{*{20}{c}} {{\xi _0} = }&\xi \end{array}} \\ {\begin{array}{*{20}{c}} {{\eta _1} = }&{{g_{10}}\xi + \eta } \end{array}} \\ {\begin{array}{*{20}{c}} {{\zeta _2} = }&{{g_{20}}\xi + {g_{21}}\eta + \zeta } \end{array}} \\ {etc} \end{array}$$

where the g_ij coefficients are implicitly defined in terms of the u_ij.