A rank-1 matrix formula applied to eigenvalue sensitivities

Abstract

This paper presents a general rank-1 matrix formula which allows for proper rearrangement of individual terms in multiproduct forms involving vectors and matrices. A far-reaching application of the new matrix formula to eigenvalue sensitivity evaluation is presented in the paper. Such an application reduces the sensitivity expressions to elegant, very fast and recursive forms with substantial savings in computer resources. The formula is applicable to rank-1 matrices of special structures which may constitute derivatives of the system state matrix, which is widely used in control system applications, with respect to various parameters of interest. In such cases, the use of the rank-1 formula yields exact non-approximate solutions which are identical to those obtained by other conventional formulas. The applicability of the rank-1 formula is believed to cover a wide variety of practical engineering systems pertaining to control and stability. © 1998 John Wiley & Sons, Ltd.