Asymptotic analysis of generalized orthogonal flows

Abstract

In this paper, we examine a matrix differential equation that approximates the k-dimensional dominant eigenspace of a matrix. We determine that its solution is orthonormal, and thus we denote this solution as the generalized orthogonal flow. We also ensure its existence and uniqueness for all time $t\in R$. In addition, we construct a particular generalized orthogonal flow that possesses minimal variation. Our findings show that the path with minimal variation is identical to an Oja-like flow. Furthermore, we conduct an in-depth analysis of the asymptotic behavior and the rate of convergence of Oja-like flow.