The term "similarity transformation" is used either to refer to a geometric similarity, or to a transformation of a square matrix into a similar matrix.
A geometric similarity preserves angles and multiplies all distances by the same positive factor. It transforms objects in space to similar objects. Geometric similarities and the concept of self-similarity are important foundations of fractals and iterated function systems.
For square matrices, a similarity transformation has the form
 |
(1)
|
where 
is an invertible matrix of the same size and
over the same field as 
. The matrices 
and 
are called similar matrices
(Golub and Van Loan 1996, p. 311). This algebraic transformation need not be
a geometric similarity.
The determinant of the similarity transformation of a matrix is equal to the determinant of the original matrix
 |  |  |
(2)
|
 |  |  |
(3)
|
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(4)
|
The determinant after subtracting a multiple of the identity matrix is given by
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
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(8)
|
If 
is an antisymmetric matrix (
) and 
is an orthogonal matrix
(
),
then the matrix for the similarity transformation
 |
(9)
|
is itself antisymmetric, i.e., 
, where 
is the transpose. This follows
using index notation for matrix multiplication,
which gives
 |  |  |
(10)
|
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(11)
|
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(12)
|
 |  |  |
(13)
|
Here, the first equality follows from the definition of matrix multiplication, and the second uses the properties of the antisymmetric
matrix 
and the orthogonal matrix 
. The third equality rearranges the factors
using the commutative property of multiplication
of scalars. The final equality follows again from the
definition of matrix multiplication.
The similarity transformation of a subgroup 
of a group 
by any fixed group element 
in 
gives the conjugate subgroup 
(Arfken 1985, p. 242).
