Matrix-Sequences of Geometric Means in the Case of Hidden (Asymptotic) Structures

In the current work, we analyze the spectral distribution of the geometric mean of two or more matrix-sequences constituted by Hermitian positive definite matrices, under the assumption that all input matrix-sequences belong to the same Generalized Locally Toeplitz (GLT) $*$-algebra. We consider the geometric mean for two matrices, using the Ando-Li-Mathias (ALM) definition, and then we pass to the extension of the the idea to more than two matrices by introducing the Karcher mean. While there is no simple formula for the geometric mean of more than two matrices, iterative methods from the literature are employed to compute it. The main novelty of the work is the extension of the study in the distributional sense when input matrix-sequences belong to one of the GLT $*$-algebras. More precisely, we show that the geometric mean of more than two positive definite GLT matrix-sequences forms a new GLT matrix-sequence, with the GLT symbol given by the geometric mean of the individual symbols. Numerical experiments are reported concerning scalar and block GLT matrix-sequences in both one-dimensional and two-dimensional cases. A section with conclusions and open problems ends the current work.