  In the recent years, the notion of rank metric in the context of coding
theory has known many interesting developments in terms of applications such as
space time coding, network coding or public key cryptography. These
applications raised the interest of the community for theoretical properties of
this type of codes, such as the hardness of decoding in rank metric. Among
classical problems associated to codes for a given metric, the notion of code
equivalence (to decide if two codes are isometric) has always been of the
greatest interest, for its cryptographic applications or its deep connexions to
the graph isomorphism problem.
  In this article, we discuss the hardness of the code equivalence problem in
rank metric for $\mathbb{F}_{q^m}$-linear and general rank metric codes. In the
$\mathbb{F}_{q^m}$-linear case, we reduce the underlying problem to another one
called {\em Matrix Codes Right Equivalence Problem}. We prove the latter
problem to be either in $\mathcal{P}$ or in $\mathcal{ZPP}$ depending of the
ground field size. This is obtained by designing an algorithm whose principal
routines are linear algebra and factoring polynomials over finite fields. It
turns out that the most difficult instances involve codes with non trivial {\em
stabilizer algebras}. The resolution of the latter case will involve tools
related to finite dimensional algebras and Wedderburn--Artin theory. It is
interesting to note that 30 years ago, an important trend in theoretical
computer science consisted to design algorithms making effective major results
of this theory. These algorithmic results turn out to be particularly useful in
the present article.
  Finally, for general matrix codes, we prove that the equivalence problem
(both left and right) is at least as hard as the well--studied {\em Monomial
Equivalence Problem} for codes endowed with the Hamming metric.
