Similar Matrices

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 .