Teoremas de Pitágoras generalizados e ângulos assimétricos entre subespaços.

Abstract

In this work, we present generalized Pythagorean theorems for multidimensional volumes in real or complex spaces. The complex case was discovered recently, and is simpler than the real one. These generalizations arise from the study of volume projection factors, which describe the contraction of volumes orthogonally projected between subspaces. To study them, we use Grasssmann esterior algebra (which deals with multivectors, or multidimensional vectors), the geometric interpretation of determinants and simple multivectors (little known in the complex case), and Jordan’s principal angles between subspaces. We also present asymmetric angles between subspaces, and show how certain trigonometric identities they satisfy are related to the generalized Pythagorean theorems, unifying the real and complex cases.