A maximal distribution result and numerical tests for geometric means of HPD GLT matrix-sequences with degenerate commuting/non-commuting GLT symbols

In the current work, we consider the study of the spectral distribution of the geometric mean of two matrix-sequences {G(An,Bn)}n formed by Hermitian Positive Definite (HPD) matrices. We assume that the two input matrix-sequences {An}n,{Bn}n belong to the same d-level r-block Generalized Locally Toeplitz (GLT) ⁎-algebra, with d,r≥1 and with GLT symbols κ,ξ. Building on recent results in the literature, we examine whether it is necessary to assume that at least one of the input GLT symbols is invertible almost everywhere. Since inversion is mainly required due to the non-commutativity of the matrix product, it was conjectured that the hypothesis on the invertibility of the GLT symbols can be removed. In fact, we prove the conjectured statement in Garoni and Serra-Capizzano (2017) [21, Conjecture 10.1], that is, the sequence of geometric means {G(An,Bn)}n is a GLT sequence whose symbol is given by the geometric mean of symbols κ and ξ (i.e., (κξ)1/2), when the symbols κ and ξ commute. This includes the important case where r=1 and d≥1. Conversely, the statement is generally false or even not well posed, when the symbols are not invertible almost everywhere and do not commute. In fact, numerical experiments are conducted in the case where the two symbols do not commute, showing that the main results of the present work are maximal. Further numerical experiments, visualizations, and conclusions complete the present contribution.