“Pandora box” — Matrix in the calculus of observations |
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| Authors: | Lubomír Kubá?ek Lea Bartalo?ová Ján Pecár Reviewer F Charamza |
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| Institution: | (1) Institute of Measurements and Measuring Equipment, Slovak Acad. Sci., Bratislava;(2) Research Institute of Geodesy and Cartography, Bratislava |
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| Abstract: | 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( ), its dispersion D(
(x)) and
2
as the unbiased estimate of the value of 
2
by means of an arbitrary g-inversion of the matrix
![$$\left {\begin{array}{*{20}c} {H,} & A \\ {A',} & O \\ \end{array} } \right]^ - = \left {\begin{array}{*{20}c} {C_1 ,} & {C_2 } \\ {C_3 } & { - C_4 } \\ \end{array} } \right]$$](/content/l3675658237u1h14/11200_2005_Article_BF01613249_TeX2GIFIE3.gif)
. The matrix
![$$\left {\begin{array}{*{20}c} {C_1 ,} & {C_2 } \\ {C_3 } & { - C_4 } \\ \end{array} } \right]$$](/content/l3675658237u1h14/11200_2005_Article_BF01613249_TeX2GIFIE4.gif)
, 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. |
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