The Constrained Rayleigh Quotient with a General Orthogonality Constraint and an Eigen-Balanced Laplacian Matrix: The Greatest Lower Bound and Applications in Cooperative Control Problems

Abstract

Mathematically, the Rayleigh quotient is defined as the quadratic function of a symmetric matrix and a nonzero (usually unconstrained) variable vector. In this paper, we consider the constrained Rayleigh quotient, in which the variable vector has the orthogonality constraint, i.e., it is constrained to be orthogonal to a nonzero vector, this nonzero vector is called the orthogonality-constraint vector (or abbreviated as the OC-vector) for the variable vector. The matrix for the Rayleigh quotient is an eigen-balanced (EB) Laplacian matrix. A tighter lower bound of the constrained Rayleigh quotient has many implications in mathematics as well as in cooperative control problems. The main contributions in this paper are as follows: First, we provide the greatest lower bound (or the infimum) of the constrained Rayleigh quotient with respect to a general OC-vector and the EB Laplacian matrix, whose results are novel and better than the existing results. Then, we interpret the physical meaning of our results in an insightful geometric form and characterize the properties of the results. Finally, as an example to illustrate the merit of the tighter lower bound of the constrained Rayleigh quotient, we consider the scale of the agents driving by a fundamental consensus protocol, and show that the convergence rate of the agents' scale is characterized by the greatest lower bound of the constrained Rayleigh quotient.