The Pythagorean theorem in certain symmetry classes of tensors

Abstract

with equality if and only if A has a zero row or A is diagonal. We are as yet unable to prove this but the subsequent inequality (3) is a step in this direction. The first purpose of this note is to exhibit (1) as a case of the Pythagorean Theorem in a suitable symmetry class of tensors. Of course, many proofs of (1) are extant and our purpose in reproving it here is to exhibit a technique that is proving itself useful for examining a wide variety of matrix functions. We then show by a similar approach that