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Chapter Three Matrix Manifolds: First-Order Geometry



The constraint sets associated with the examples discussed in Chapter 2 have a particularly rich geometric structure that provides the motivation for this book. The constraint sets are matrix manifolds in the sense that they are manifolds in the meaning of classical differential geometry, for which there is a natural representation of elements in the form of matrix arrays. The matrix representation of the elements is a key property that allows one to provide a natural development of differential geometry in a matrix algebra formulation. The goal of this chapter is to introduce the fundamental concepts in this direction: manifold structure, tangent spaces, cost functions, differentiation, Riemannian metrics, and gradient computation. There are two classes of matrix manifolds that we consider in detail in this book: embedded submanifolds of R n×p and quotient manifolds of R n×p (for 1 ≤ p ≤ n). Embedded submanifolds are the easiest to understand, as they have the natural form of an explicit constraint set in matrix space R n×p. The case we will be mostly interested in is the set of orthonormal n ×

Year: 2010
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