Fixed-Rank Manifold: Tangent Spaces and Local Defining Functions

This lecture delves into the concept of fixed-rank matrices, exploring how to show that they form an embedded submanifold. The instructor explains the challenges in constructing local defining functions for the manifold and demonstrates how to determine the dimension and tangent spaces of the manifold using a practical approach. By representing matrices in a specific format, the lecture showcases how to efficiently compute tangent vectors at a given point on the manifold. The discussion emphasizes the importance of this approach in managing memory and computational complexity when working with high-dimensional matrices.