This paper introduces new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds, which are used to enforce orthogonality constraints in numerical linear algebra. These manifolds arise in various applications such as symmetric eigenvalue problems, nonlinear eigenvalue problems, electronic structures computations, and signal processing. The paper presents a geometric framework that unifies seemingly unrelated algorithms and provides a top-level mathematical view of these algorithms. The framework is based on differential geometry and numerical linear algebra, and it connects ideas from different areas such as optimization, signal processing, and electronic structures computations. The paper also provides a taxonomy for numerical linear algebra algorithms and suggests new algorithms. The key contributions include new algorithms for Newton and conjugate gradient methods on the Grassmann and Stiefel manifolds, a geometrical framework for algorithms involving orthogonality constraints, and a discussion of the geometry of the Grassmann and Stiefel manifolds. The paper also discusses the canonical metric on these manifolds and provides formulas for geodesics and parallel translation. The paper concludes with a discussion of the computational complexity of these algorithms and their potential applications in various fields.This paper introduces new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds, which are used to enforce orthogonality constraints in numerical linear algebra. These manifolds arise in various applications such as symmetric eigenvalue problems, nonlinear eigenvalue problems, electronic structures computations, and signal processing. The paper presents a geometric framework that unifies seemingly unrelated algorithms and provides a top-level mathematical view of these algorithms. The framework is based on differential geometry and numerical linear algebra, and it connects ideas from different areas such as optimization, signal processing, and electronic structures computations. The paper also provides a taxonomy for numerical linear algebra algorithms and suggests new algorithms. The key contributions include new algorithms for Newton and conjugate gradient methods on the Grassmann and Stiefel manifolds, a geometrical framework for algorithms involving orthogonality constraints, and a discussion of the geometry of the Grassmann and Stiefel manifolds. The paper also discusses the canonical metric on these manifolds and provides formulas for geodesics and parallel translation. The paper concludes with a discussion of the computational complexity of these algorithms and their potential applications in various fields.