Jordan normal form
Every complex square matrix is similar to a block matrix of Jordan blocks ; it is diagonalisable exactly when every Jordan block is .
Any complex matrix is similar to a matrix with block form given by
where each Jordan block is a matrix of the form
with , and are the eigenvalues of and (because they are similar).
Note that the same eigenvalue may appear in multiple Jordan blocks.
is diagonalisable iff all Jordan blocks are of size .
Two-dimensional case
Any complex matrix is similar to one of:
- with ;
- ;
- .
In case 1 the eigenvectors form a basis and is diagonal. In case 2 the same holds with a repeated eigenvalue of full geometric multiplicity. In case 3, where but , take an eigenvector and extend it to a basis ; then with , and replacing by makes the matrix of the map exactly .
Related
Stated in
- Theorem 4.30 (General Jordan Normal Form)§4.8 Jordan Normal Forms