Similar matrices are two square matrices 
and 
that are related by
 |
(1)
|
where 
is a square nonsingular matrix over the same
field. A transformation of the form 
is called a similarity
transformation, or conjugation by 
. For example,
![[0 1; 0 0]](https://mathworld.wolfram.com/images/equations/SimilarMatrices/NumberedEquation2.svg) |
(2)
|
and
![[0 0; 1 0]](https://mathworld.wolfram.com/images/equations/SimilarMatrices/NumberedEquation3.svg) |
(3)
|
are similar under conjugation by
![C=[0 1; 1 0].](https://mathworld.wolfram.com/images/equations/SimilarMatrices/NumberedEquation4.svg) |
(4)
|
Similar matrices represent the same linear transformation after a change of basis (for the domain
and range simultaneously). Recall that a matrix
corresponds to a linear transformation,
and a linear transformation corresponds
to a matrix after choosing a basis 
,
 |
(5)
|
Changing the basis changes the coefficients of the matrix,
 |
(6)
|
If 
uses the standard basis vectors,
then 
is the matrix 
using the basis vectors 
.
