Eigenvalue and state-transition sensitivity of linear systems

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Abstract
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.

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