Similar matrices

Square matrices $A$ and $B$ of the same order related by $B=S^{-1}AS$, where $S$ is a non-degenerate matrix of the same order. Similar matrices have the same rank, the same determinant, the same characteristic polynomial, and the same eigenvalues. It is often important to select a matrix similar to a given one but having a possibly simpler form, for example, diagonal form (see Diagonal matrix) or Jordan form (see Jordan matrix).

Comments

Similar matrices arise when an endomorphism of a finite-dimensional vector space over a field (a linear map of the space to itself) is represented by matrices $A$, $B$ with respect to two different bases, the change of basis being expressed by the matrix $S$. The rank, determinant, trace, characteristic polynomial and so forth are properties of the endomorphism.

Similarity is an equivalence relation on matrices. Over an algebraically closed field, the Jordan matrix provides a canonical representative of each similarity class.

References

[a1]	Paul R. Halmos, Finite-dimensional vector spaces, Undergraduate texts in mathematics, Springer (1974)