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)