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A simple formula is proved to be a tight estimate for the condition number ofthe full rank linear least squares residual with respect to the matrix of leastsquares coefficients and scaled 2-norms. The tight estimate reveals that thecondition number depends on three quantities, two of which can causeill-conditioning. The numerical linear algebra literature presents severalestimates of various instances of these condition numbers. All the prior valuesexceed the formula introduced here, sometimes by large factors.

Spectral Condition Numbers of Orthogonal Projections and Full Rank Linear Least Squares Residuals