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In Hoffman & Kunze, it is mentioned several times that any matrix with a minimal polynomial that splits over your field into linear factors is similar to an upper triangular matrix. A proof is given, but it does not give an algorithm for finding such a matrix. Indeed, they prove that any such linear operator can be decomposed into the sum of a nilpotent linear operator and a diagonalizable linear operator which commute, and can therefore be simultaneously triangularized, but do not seem to give a direct method as to how to actually choose an appropriate basis in practice.

I was wondering if there is an algorithm for triangularizing such an operator that does not rely on Jordan canonical form (since triangularization appears before Jordan form in their book) or noticing some clever choice of basis?

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The upper triangular matrix is simply the representation with respect to a basis of generalized eigenvectors for the roots (eigenvalues) appearing in the minimal polynomial (assumed to split into linear factors). The details for extracting the generalized eigenvectors may be somewhat tedious, but straightforward. Consider the case of a single eigenvalue, and specialize it to zero for clarity. –  hardmath Aug 4 '12 at 15:08
    
Use induction on the size of the matrices. –  i. m. soloveichik Aug 4 '12 at 15:09

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Take an eigenvector $v$. Consider the transformation on $W=V/Fv$ (field is $F$). The factorization for minimal polynomial on $W$ is still valid so transformation is upper triangular on $W$. Then lift this back to $V$ to get upper triangular form on $V$.

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