On the geometrical structure of symmetric matrices

The identity ray, αI for α>0, can be seen as the center ray of the cone of symmetric and positive definite (SPD) matrices. In that sense, the angle that any SPD matrix forms with the identity plays a very important role to understand the geometrical structure of the cone. In this work, we extend this relationship, and analyze the geometrical structure of symmetric matrices including the location of all orthogonal matrices, not only the identity matrix. This geometrical understanding leads to new results in the subspace of symmetric matrices. We also extend some of the geometrical results for the case of general (not necessarily symmetric) nonsingular matrices.