Change of Basis for Matrices
Matrices representing the same linear map in different bases are related by for some .
Proposition 9.5 (Change of Basis)
Let be an -dimensional vector space over , and a linear map. If that represents in some basis, then the orbit
consists of all matrices that represent in any basis.
Proof
A basis defines an isomorphism . If represents in this basis, then . For another basis with coordinate isomorphism , a matrix represents when .
Therefore . Writing for the matrix of in the standard basis gives . Conversely, setting constructs a basis in which this conjugate matrix represents .
Related
Stated in
- Proposition 9.5 (Change of Basis)§9.1 Change of Basis