Some matrix functional equations

We investigate the pair of matrix functional equations $\mathbf G(x)\mathbf F(y)= \mathbf G(xy)$ and $\mathbf G(x)\mathbf G(y)=\mathbf F(y/x)$, featuring the two independent scalar variables $x$ and $y$ and the two $N\times N$ matrices $\mathbf F(z)$ and $\mathbf G(z)$ (with $N$ an arbitrary positive integer and the elements of these two matrices functions of the scalar variable $z$). We focus on the simplest class of solutions, i.e., on matrices all of whose elements are analytic functions of the independent variable. While in the scalar $(N=1)$ case this pair of functional equations only possess altogether trivial constant solutions, in the matrix $(N>1)$ case there are nontrivial solutions. The se solutions satisfy the additional pair of functional equations $\mathbf F(x)\mathbf G(y)=\mathbf G(y/x)$ and $\mathbf F(x)\mathbf F(y) =\mathbf F(xy)$, and an endless hierarchy of other functional equations featuring more than two independent variables.