# Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective

## TL;DR

This paper investigates orthogonal tensor decompositions over real and complex fields using linear algebra, providing new proofs, explicit descriptions, and initial insights into the closures of decomposable tensor sets.

## Contribution

It offers a new proof for real orthogonal tensor decompositions, describes complex decomposable tensors with polynomial conditions, and initiates the study of their closures.

## Key findings

- Decomposable tensors over reals are defined by quadratic equations.
- Explicit polynomial descriptions for complex decomposable tensors are provided.
- Partial results on tensor closure characterization are presented, linked to approximate diagonalization.

## Abstract

We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined be equations of degree 2. This gives a new proof of some of the results of Robeva and Boralevi et al. Orthogonal decompositions over the field of complex numbers had not been studied previously; we give an explicit description of the set of decomposable tensors using polynomial equalities and inequalities, and we begin a study of their closures. The main open problem that arises from this work is to obtain a complete description of the closures. This question is akin to that of characterizing border rank of tensors in algebraic complexity. We give partial results using in particular a connection with approximate simultaneous diagonalization (the so-called "ASD property").