Two square matrices and that are related by
(1)
|
where is a square nonsingular matrix are said to be similar. A transformation of the form is called a similarity transformation, or conjugation by . For example,
(2)
|
and
(3)
|
are similar under conjugation by
(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 .