  This paper is a report based on the results obtained during a three months
internship at the University of Pittsburgh by the first author and under the
mentorship of the second author. The notion of an amoeba of a subvariety in a
torus $(\mathbb{C}^*)^n$ has been extended to subvarieties of the general
linear group $GL_n(\mathbb{C})$ by the second author and Manon. In this paper,
we show some basic properties of these matrix amoebas, e.g. any such amoeba is
closed and the connected components of its complement are convex when the
variety is a hypersurface. We also extend the notion of Ronkin function to this
setting. For hypersurfaces, we show how to describe the asymptotic directions
of the matrix amoebas using a notion of Newton polytope. Finally, we partially
extend the classical statement that the amoebas converge to the tropical
variety. We also discuss a few examples. Our matrix amoeba should be considered
as the Archimedean version of the spherical tropicalization of
Tevelev-Vogiannou for the variety $GL_n(\mathbb{C})$ regarded as a spherical
homogeneous space for the left-right action of $GL_n(\mathbb{C}) \times
GL_n(\mathbb{C})$.
