A state-of-the-art review on theory and engineering applications of eigenvalue and eigenvector derivatives

•The complete range of real, complex, fractional, linear and nonlinear eigenproblems is considered.•Formulae are developed for the eigen-derivatives of distinct, repeated, non-self-adjoint and defective eigenvalue problems.•In depth treatment of theory and applications is given.•Opportunities for further research are revealed.•Up-to-date references are provided. Eigenvalue and eigenvector derivatives with respect to system design variables and their applications have been and continue to be one of the core issues in the design, control and identification of practical engineering systems. Many different numerical methods have been developed to compute accurately and efficiently these required derivatives from which, a wide range of successful applications have been established. This paper reviews and examines these methods of computing eigenderivatives for undamped, viscously damped, nonviscously damped, fractional and nonlinear vibration systems, as well as defective systems, for both distinct and repeated eigenvalues. The underlying mathematical relationships among these methods are discussed, together with new theoretical developments. Major important applications of eigenderivatives to finite element model updating, structural design and modification prediction, performance optimization of structures and systems, optimal control system design, damage detection and fault diagnosis, as well as turbine bladed disk vibrations are examined. Existing difficulties are identified and measures are proposed to rectify them. Various examples are given to demonstrate the key theoretical concepts and major practical applications of concern. Potential further research challenges are identified with the purpose of concentrating future research effort in the most fruitful directions.