“Pandora box” — Matrix in the calculus of observations

Summary

If the condition R(A)=k(≦n), whereA is the design matrix of the type n × k and k the number of parameters to be determined, is not satisfied, or if the covariance matrixH is singular, it is possible to determine the adjusted value of the unbiased estimable function of the parameters f(Θ), its dispersion D($\hat f$(x)) and$\hat \sigma$ ² as the unbiased estimate of the value of σ² by means of an arbitrary g-inversion of the matrix$\left[ {\begin{array}{*{20}c} {H,} & A \\ {A',} & O \\ \end{array} } \right]^ - = \left[ {\begin{array}{*{20}c} {C_1 ,} & {C_2 } \\ {C_3 } & { - C_4 } \\ \end{array} } \right]$. The matrix$\left[ {\begin{array}{*{20}c} {C_1 ,} & {C_2 } \\ {C_3 } & { - C_4 } \\ \end{array} } \right]$, because of its remarkable properties, is called the “Pandora Box” matrix. The paper gives the proofs of these properties and the manner in which they can be employed in the calculus of observations.