Chapter Three - Rank Deficiency and Ill-Conditioning

Abstract

The characteristics of rank-deficient and ill-conditioned linear systems of equations are explored using the singular value decomposition. The connection between model and data null spaces and solution uniqueness and ability to fit data is examined. Model and data resolution matrices are defined. The relationship between singular value size and singular vector roughness and its connection to the effect of noise on solutions are discussed in the context of the fundamental trade-off between model resolution and instability. Specific manifestations of these issues in rank-deficient and ill-conditioned discrete problems are shown in several examples.