Abstract
In mathematics and engineering, a manifold is a topological space that locally resembles Euclidean space near each point. Defining the best metric for these manifolds have several engineering and science implications from controls to optimization for generalized inner product applications of Gram Matrices that appear in these applications. These smooth geometric manifold applications can be formalized by Lie Groups and their Lie Algebras on its infinitesimal elements. Nilpotent matrices that are matrices with zero power with left-invariant metric on Lie group with non-commutative properties namely non-abelian nilsoliton metric Lie algebras will be the focus of this article. In this study, we present an algorithm to classify eigenvalues of nilsoliton derivations for 9-D non-abelian nilsoliton metric Lie algebras with non-singular Gram matrices.