The term "similarity transformation" is used either to refer to a geometric similarity, or to a matrix transformation that results in a similarity.
A similarity transformation is a conformal mapping whose transformation matrix 
can be written in the form
 |
(1)
|
where 
and 
are called similar matrices (Golub and Van Loan
1996, p. 311). Similarity transformations transform objects in space to similar
objects. Similarity transformations and the concept of self-similarity
are important foundations of fractals and iterated
function systems.
The determinant of the similarity transformation of a matrix is equal to the determinant of the original matrix
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
The determinant of a similarity transformation minus a multiple of the unit matrix is given by
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
If 
is an antisymmetric matrix (
) and 
is an orthogonal matrix
(
),
then the matrix for the similarity transformation
 |
(9)
|
is itself antisymmetric, i.e., 
. This follows using index notation for matrix
multiplication, which gives
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
Here, equation (10) follows from the definition of matrix multiplication, (11) uses the properties of antisymmetry
in 
and orthogonality in 
, (12) is a rearrangement of (11)
allowed since scalar multiplication is commutative, and (13)
follows again from the definition of matrix
multiplication.
The similarity transformation of a subgroup 
of a group 
by a fixed element 
in 
not in 
always gives a subgroup (Arfken
1985, p. 242).
