Conditioning and data normalization

The condition number may depend on the numerical representation of the problem. An appropriate choice of measurement units and normalization of the variables can reduce scale imbalances among the different components of the matrix and improve the numerical behavior of the solution.

This does not mean, however, that simple rescaling eliminates the intrinsic sensitivity of the problem: the choice of parameterization changes the representation of the variables and must be evaluated in relation to the physical and geometric problem under consideration.

Conditioning therefore provides a measure of the deterministic sensitivity of the solution to perturbations in the data. When perturbations are instead modeled as random variables characterized by a distribution and a covariance matrix, it is possible to describe not only the magnitude of the sensitivity, but also how statistical uncertainty is transferred from the measurement to the solution. This second perspective will be introduced in section 2.6.