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The condition number of solutions to full rank linear least-squares problemare shown to be given by an optimization problem that involves nuclear norms ofrank 2 matrices. The condition number is with respect to the least-squarescoefficient matrix and 2-norms. It depends on three quantities each of whichcan contribute ill-conditioning. The literature presents several estimates forthis condition number with varying results; even standard reference textscontain serious overestimates. The use of the nuclear norm affords a singlederivation of the best known lower and upper bounds on the condition number andshows why there is unlikely to be a closed formula.

Nuclear Norms of Rank 2 Matrices for Spectral Condition Numbers of Full Rank Linear Least Squares Solutions