We will learn how to see the space of matrices of a given size from a geometric and topological perspective, with emphasis on the families of various ranks and how they fit together.  We pay special attention to the nearest orthogonal neighbor and nearest singular neighbor of a given matrix, because both play central roles in matrix decompositions.  We will give applications to facial recognition protocols and to the study of interest rate term structures for US treasury bonds.  The talk describes joint work with five graduate students: Amora Elsaify (PhD student in Finance at Penn), Ben Grossmann (PhD student in Math at Drexel), Joe Hoisington, Anusha Krishnan and Jianru Zhang (the last three PhD students in Math at Penn).