# Classification of linear mappings between indefinite inner product spaces

## TL;DR

This paper provides a comprehensive classification of linear mappings between indefinite inner product spaces over real, complex, and quaternionic fields, considering various forms and field characteristics.

## Contribution

It offers a complete classification of triples involving linear maps and symmetric, skew-symmetric, or Hermitian forms over multiple fields, extending existing results.

## Key findings

- Classification over real, complex, and quaternionic fields
- Inclusion of symmetric, skew-symmetric, and Hermitian forms
- Results applicable when characteristic of field is not 2

## Abstract

Let $\mathcal A:U\to V$ be a linear mapping between vector spaces $U$ and $V$ over a field or skew field $\mathbb F$ with symmetric, or skew-symmetric, or Hermitian forms $\mathcal B:U\times U\to\mathbb F$ and $\mathcal C:V\times V\to\mathbb F.$ We classify the triples $(\mathcal A,\mathcal B,\mathcal C)$ if $\mathbb F$ is $\mathbb R$, or $\mathbb C$, or the skew field of quaternions $\mathbb H$. We also classify the triples $(\mathcal A,\mathcal B,\mathcal C)$ up to classification of symmetric forms and Hermitian forms if the characteristic of $\mathbb F$ is not 2.