How SPD Matrices Encode Inner Products

Symmetric Positive Definite (SPD) matrices and inner products are central concepts in linear algebra and vector space geometry. In this video, we explain how SPD matrices define inner products in finite-dimensional real vector spaces and why they are essential for understanding geometric structure. We show that every inner product corresponds to an SPD matrix in a given coordinate system, and how changing this matrix leads to a new inner product. This means we can redefine distances, angles, and geometry within the same vector space simply by choosing a different SPD matrix. By the end of this video, you will understand how SPD matrices connect algebra and geometry, and how they allow us to view a single vector space through multiple geometric perspectives.