Matrix consimilarity

In linear algebra, two "n"-by-"n" matrices "A" and "B" are called consimilar if
for some invertible formula_2 matrix formula_3, where formula_4 denotes the elementwise complex conjugation. So for real matrices similar by some real matrix formula_3, consimilarity is the same as matrix similarity. 
Like ordinary similarity, consimilarity is an equivalence relation on the set of formula_2 matrices, and it is reasonable to ask what properties it preserves. 
The theory of ordinary similarity arises as a result of studying linear transformations referred to different bases. Consimilarity arises as a result of studying antilinear transformations referred to different bases. 
A matrix is consimilar to itself, its complex conjugate, its transpose and its adjoint matrix. Every matrix is consimilar to a real matrix and to a Hermitian matrix. There is a standard form for the consimilarity class, analogous to the Jordan normal form.