algorithm: diagonalize a matrix

Let be a matrix.
Suppose is diagonalizable (cf. criterias: diagonalizable Matrix).

Then to diagonalize it, one has to find an invertible matrix and a diagonal matrix such that

1. determine the eigenvalues - if you only have to find the diagonal form, then you are done:
   

Set

where are the eigenvalues

1. determine the eigenspace
   
2. find a basis consisting of eigenvectors: find a basis for each Eigenspace. Then a Basis of is given by taking the union of the basis
   
3. Suppose form a basis consisting of eigenvectors (regarded as column vector), where is the eigenvalue for . Then consider the matrix
   

as basis change from to the standard basis
Then we get

and

Hence to determine it remains to compute the inverse of