Matrix Ill-conditioning

Matrix ill-conditioning is a general term used to describe a rectangular matrix of values which is unsuitable for use in a particular analysis.
This occurs perhaps most frequently in applications of linear multiple regression when the matrix of correlations for the predictors is singular and thus the regular matrix inverse cannot be computed. In some modules (i.e., in Factor Analysis) this problem is dealt with by issuing a respective warning and then artificially lowering all correlations in the correlation matrix by adding a small constant to the diagonal elements of the matrix, and then restandardizing it. This procedure will usually yield a matrix for which the regular matrix inverse can be computed. Note that in many applications of the general linear model and the generalized linear model, matrix singularity is not abnormal (i.e., when the overparameterized model is used to represent effects for categorical predictor variables) and is dealt with by computing a generalized inverse rather than the regular matrix inverse. Another example of matrix ill-conditioning is intransitivity of the correlations in a correlation matrix. If in a correlation matrix variable A is highly positively correlated with B, B is highly positively correlated with C, and A is highly negatively correlated with C, this "impossible" pattern of correlations signals an error in the elements of the matrix.
See also matrix singularity, matrix inverse, generalized inverse.