A note on adjustment of free networks

Abstract

The present paper deals with the least-squares adjustment where the design matrix (A) is rank-deficient. The adjusted parameters\(\hat x\) as well as their variance-covariance matrix (\(\sum _{\hat x} \)) can be obtained as in the “standard” adjustment whereA has the full column rank, supplemented with constraints,\(C\hat x = w\), whereC is the constraint matrix andw is sometimes called the “constant vector”. In this analysis only the inner adjustment constraints are considered, whereC has the full row rank equal to the rank deficiency ofA, andAC T =0. Perhaps the most important outcome points to the three kinds of results 1) A general least-squares solution where both\(\hat x\) and\(\sum _{\hat x} \) are indeterminate corresponds tow=arbitrary random vector. 2) The minimum trace (least-squares) solution where\(\hat x\) is indeterminate but\(\sum _{\hat x} \) is detemined (and trace\(\sum _{\hat x} \) corresponds tow=arbitrary constant vector. 3) The minimum norm (least-squares) solution where both\(\hat x\) and\(\sum _{\hat x} \) are determined (and norm\(\hat x\), trace\(\sum _{\hat x} \) corresponds tow−0