In linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, two n-by-n matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 A and B are called consimilar if


for some invertible matrix , where denotes the elementwise complex conjugation. So for real matrices similar by some real matrix , consimilarity is the same as similarity.

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

The theory of ordinary similarity arises as a result of studying linear transformation

Linear transformation

In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

s referred to different bases. Consimilarity arises as a result of studying antilinear transformations referred to different bases.