In linear algebra, two n-by-n matrices A and B are called consimilar if
A=SB\bar{S}-1
for some invertible
n x n
S
\bar{S}
S
Like ordinary similarity, consimilarity is an equivalence relation on the set of
n x n
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.