2023/08/11 by Octavio Arizmendi, Arizmendi, Octavio, Franz Lehner +3 · 1 citation
Computer Science · Mathematics · #46L54 #60B20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Polynomial and algebraic computation #Probability (math.PR) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2309.14343
openalex publication_date 2023/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study pairs (A,B) of square matrices that are in additive (resp. multiplicative) finite free position, that is, the characteristic polynomial χA+B(x) (resp. χAB(x)) equals the additive finite free convolution χA(x) \boxplus χB(x) (resp. the multiplicative finite free convolution χA(x) \boxtimes χB(x)), which equals the expected characteristic polynomial 𝔼U [ χA+U^* BU(x) ] (resp. 𝔼U [ χAU^* BU(x) ]) over the set of unitary matrices U. We examine the lattice of (non-irreducible) affine algebraic sets of matrices consisting of finite free complementary pairs with respect to the additive (resp. multiplicative) convolution. We show that these pairs include the diagonal matrices vs. the principally balanced matrices, the upper (lower) triangular matrices vs. the upper (lower) triangular matrices with constant diagonal, and the scalar matrices vs. the set of all square matrices.