Consimilar matrix

In linear algebra, two n-by-n matrices A and B are called consimilar if

$A = S B \bar{S}^{-1} \,$

for some invertible $n \times n$ matrix $S$ , where $\bar{S}$ denotes the elementwise complex conjugation. So for real matrices similar by some real matrix $S$ , consimilarity is the same as similarity.

Like ordinary similarity, consimilarity is an equivalence relation on the set of $n \times n$ matrices, and it is reasonable to ask what properties it preserve.

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.

References

Worn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2. (section 4.5 and 4.6 discusses consimilarity)