We have already seen, in chapter 2, that if A is square and nonsingular a unique solution of the equation Ax = b exists for any arbitrary b. Equations of this form arise frequently when analysing the static behaviour of physical and economics systems and often represent the response of the system to the particular set of applied stimuli embodied in the vector b. If, however, we wish to investigate the dynamic behaviour of such systems we are led to the determination of those values of the scalar λ for which A — λI is singular. These values of λ are associated with the innate properties of the system under investigation, for example, the natural frequencies of vibration of mechanical or electrical systems or certain variances obtained in statistical analyses, and as such are independent of applied stimuli. This is in marked contrast to the solution of the related set of linear equations Ax = b.