The geometry of mixed-Euclidean metrics on symmetric positive definite matrices

Several Riemannian metrics and families of were defined on the manifold Symmetric Positive Definite (SPD) matrices. Firstly, we formalize a common general process to define metrics: principle deformed metrics. We relate recently introduced family alpha-Procrustes class mean kernel by providing sufficient condition under which elements former belong latter. Secondly, focus balanced bilinear forms that introduced. give new form is metric. It allows us introduce Mixed-Euclidean (ME) generalize Mixed-Power-Euclidean (MPE) unveal their link with (u,v)-divergences (α,β)-divergences information geometry provide an explicit formula Riemann curvature tensor. show sectional all ME can take negative values experimentally MPE but log-Euclidean, power-Euclidean power-affine positive values.