Definition: Similar matrices.
is called similar to if there exists an invertible matrix such that .
- We can think of
as the same linear transformation written in two different bases is the matrix that converts coordinates from one basis to another
- When
are similar they share the following: - determinant
- eigenvalues
- characteristic polynomial
- rank
Proof: Similar matrices have the same determinant and characteristic polynomial Let
be some linear transformation. A similar matrix can be defined as followed.