Simultaneous orthogonalization of inner products over arbitrary fields

TL;DR

This paper establishes the precise conditions under which a set of inner products on a finite-dimensional vector space over any field can share an orthogonal basis, with applications to evolution algebras.

Contribution

It provides necessary and sufficient criteria for simultaneous orthogonalization of multiple inner products over arbitrary fields, extending classical results beyond real and complex fields.

Findings

Characterization of when a family of inner products admits a common orthogonal basis

Extension of orthogonalization conditions to arbitrary fields

Applications to the structure of evolution algebras

Abstract

We give necessary and sufficient conditions for a family of inner products in a finite-dimensional vector space $V$ over an arbitrary field $\mathbb{K}$ to have an orthogonal basis relative to all the inner products. Some applications to evolution algebras are also considered.