Statistical Methods
Assess sensitivity before trusting a two-variable linear solve
Define the task
A linear-system solution can have a small residual while remaining sensitive to small changes in the inputs. Record matrix scaling and the norm used to report its condition number. For the spectral condition number, divide the largest singular value by the smallest. A singular matrix has no finite value under this definition.
Carry out the check
Test both the actual coefficient matrix and a deliberately perturbed right-hand side. Relative forward-error bounds depend on conditioning and on the type and size of perturbation. Treat a condition number as a sensitivity diagnostic. It is not an automatic count of correct decimal places or a statistical goodness-of-fit score.
Worked hypothetical example
A hypothetical diagonal matrix with entries 1 and 0.001 has spectral condition number 1000. Solving against a right-hand side of (1,0) gives (1,0). Changing only the second right-hand component to 0.001 changes the solution to (1,1). Multiplying the entire matrix by a nonzero constant leaves the condition number unchanged. Rescaling only one coordinate can change it, so preserve the unit convention in any comparison.
Method checked 12 November 2026 against R documentation: Compute or Estimate the Condition Number of a Matrix.