Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The sum and the product of two quadratic matrices   Let $p$ and $q$ be polynomials with degree $2$ over an arbitrary field
$\mathbb{F}$.
  In the first part of this article, we characterize the matrices that can be
decomposed as $A+B$ for some pair $(A,B)$ of square matrices such that $p(A)=0$
and $q(B)=0$. The case when both polynomials $p$ and $q$ are split was already
known. In the first half of this article, we complete the study by tackling the
case when at least one of the polynomials $p$ and $q$ is irreducible over
$\mathbb{F}$.
  In the second half of the article, we use a similar method to characterize,
under the assumption that $p(0)q(0) \neq 0$, the matrices that can be
decomposed as $AB$ for some pair $(A,B)$ of square matrices such that $p(A)=0$
and $q(B)=0$.

16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph