标题： HPD GLT矩阵序列的几何平均：超越GLT符号可逆性假设的最大结果 显示英文标题

标题： Geometric means of HPD GLT matrix-sequences: a maximal result beyond invertibility assumptions on the GLT symbols

摘要： 在本研究中，我们研究由厄米特正定（HPD）矩阵构成的两个矩阵序列 $\{G(A_n, B_n)\}_n$ 的几何平均矩阵序列的谱分布。假设两个输入矩阵序列 $\{A_n\}_n, \{B_n\}_n$ 属于同一个 $d$ 层 $r$ 块广义局部托普利茨 (GLT) $\ast$ 代数，其中 $d,r\ge 1$ 且 GLT 符号为 $\kappa, \xi$。 基于文献中的最新成果，我们检验了至少有一个输入 GLT 符号几乎处处可逆（a.e.）的假设是否必要。 由于矩阵乘积的非交换性主要导致需要进行逆运算，因此推测可以取消关于 GLT 符号可逆性的假设。 事实上，我们证明了当符号 $\kappa, \xi$ 可交换时，\[\{G(A_n, B_n)\}_n \sim_{\mathrm{GLT}} (\kappa \xi)^{1/2} \] 的猜想，这蕴含了 $r=1$ 和 $d \geq 1 $ 的重要情形。而当符号不可逆且不可交换时，该命题通常为假，甚至不成立。 事实上，在两个符号不可交换的情况下进行了数值实验，结果表明本研究的主要结果是最优的。 进一步的数值实验、可视化和结论结束了本研究。 显示英文摘要

摘要： In the current work, we consider the study of the spectral distribution of the geometric mean matrix-sequence of two matrix-sequences $\{G(A_n, B_n)\}_n$ formed by Hermitian Positive Definite (HPD) matrices, assuming that the two input matrix-sequences $\{A_n\}_n, \{B_n\}_n$ belong to the same $d$-level $r$-block Generalized Locally Toeplitz (GLT) $\ast$-algebra with $d,r\ge 1$ and with GLT symbols $\kappa, \xi$. Building on recent results in the literature, we examine whether the assumption that at least one of the input GLT symbols is invertible almost everywhere (a.e.) is necessary. 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 that is \[ \{G(A_n, B_n)\}_n \sim_{\mathrm{GLT}} (\kappa \xi)^{1/2} \] when the symbols $\kappa, \xi$ commute, which implies the important case where $r=1$ and $d \geq 1 $, while the statement is generally false or even not well posed when the symbols are not invertible a.e. 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 end the present contribution.