Eigenvalue and state-transition sensitivity of linear systems

Differential changes in the elements of a matrix associated with a linear multivariable dynamic system produce changes in the corresponding eigenvalues and in the state-variable solution. Previous methods for determining the eigenvalue sensitivity are outlined, and an alternative development based on Sylvester´s expansion theorem is discussed which illustrates the basic role of the constituent matrices associated with the theory of linear systems. Methods for determining the corresponding variations in the transition- and driving-matrix elements related to the time response of linear systems are also illustrated. The inverse eigenvalue sensitivity problem concerned with the requirement to synthetise a differential change in the elements of a matrix to produce a desired eigenvalue change is also considered. A numerical procedure is proposed, together with a solution based on the generalised inverse of a matrix, for solving the defining singular equations.