A Determinant Proof of a Generalized Pythagorean Theorem

Abstract

A generalized Pythagorean theorem is an equation relating the squares of the volumes of faces of a particular k-simplex in n-dimensional Euclidean space. There are many proofs of this theorem. This note presents yet another very short elementary proof of the generalized Pythagorean theorem that is no more than the direct expansion of the determinant of A^TA where A is a matrix determined by the simplex.

• MSC: Primary 51A15
• Secondary 51A04
• 51A05

Acknowledgments

This note is dedicated to Professor Richard A. Brualdi at the University of Wisconsin-Madison on the occasion of his 80th birthday. The authors thank the reviewers and the editor for their valuable comments.