The sum of two quadratic matrices: Exceptional cases

Let p and q be polynomials with degree 2 over an arbitrary field F. A square matrix with entries in F is called a (p, q) -sum when it can be split into A + B for some pair (A, B) of square matrices such that p(A) = 0 and q(B) = 0. A (p, q)-sum is called regular when none of its eigenvalues is the sum of a root of p and of a root of q. A (p, q)-sum is called exceptional when each one of its eigenvalues is the sum of a root of p and of a root of q. In a previous work [7], we have shown that the study of (p, q)-sums can be entirely reduced to the one of regular (p, q)-sums and to the one of exceptional (p, q)-sums. Moreover, regular (p, q)-sums have been characterized thanks to structural theorems on quaternion algebras, giving the problem a completely unified treatment. The present article completes the study of (p, q)-sums by characterizing the exceptional ones. The new results here deal with the case where at least one of the polynomials p and q is irreducible over F. (c) 2022 Elsevier Inc. All rights reserved.