RANK ONE PERTURBATIONS AND APPLICATIONS

Yueqiao Zhang

The role of rank one perturbations in transforming the eigenstructure of a matrix has long been considered in the context of applications, especially in linear control systems. Three cases are examined as part of this work: First, we propose a practical method to place the system eigenvalues in any desired locations for a system that is completely controllable via an appropriate choice of feedback control, found via sequential rank one perturbations. Second, we stabilize a system that is stabilizable but not neccessarily completely controllable by placing the system eigenvalues in the open left-half complex plane through feedback via rank one perturbations. Third, a choice of feedback control is proposed in order to achieve that a trajectory of a linear control system eventually enters an orthant of Rn and remains therein for all time thereafter. The last situation is achieved by imposing the strong Perron-Frobenius property and involves altering an eigenvalue as well as its corresponding eigenvector appropriately. P-matrices have positive principal minors and include many well-known matrix classes (positive definite, totally positive, M-matrices etc.) How does one construct a generic P-matrix? Specifically, is there a characterization of P-matrices that lends itself to the tractable construction of every P-matrix? To answer these questions positively, a recursive method is employed that is based on a characterization of rank-one perturbations that preserve the class of P-matrices. Nonnegative matrices have long been a source of interesting and challenging mathematical problems. They are real matrices with all their entries being nonnegative and arise in a number of important application areas: communications systems, biological systems, economics, ecology, computer sciences, machine learning, and many other engineering systems. We explore applications of rank one perturbations on the nonnegative inverse eigenvalue problem as well as on stochastic matrices.

RANK ONE PERTURBATIONS AND APPLICATIONS

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