# finite steps to Hessenberg form and/or triangular form

I am learning numerical linear algebra and curious about one thing. It is possible to reduce any matrix to the Hessenberg form in finite steps with a unitary matrix. But why is it impossible to reduce it further into an upper triangular form in finite step?

What is the fundamental barrier?

If you could reduce to a triangular matrix $$A = QTQ^*$$ (a Schur factorization) in a finite number of steps (involving elementary arithmetic operations and n-th roots only), this would violate the Abel-Ruffini theorem: it would allow you to exactly (neglecting roundoff errors) compute roots of arbitrary polynomials in a finite number of steps.
The reason is that the diagonal entries of the triangular matrix $$T$$ are equal to the eigenvalues of $$A$$, and for any degree-$$n$$ polynomial $$p(z)$$ you can find the roots by transforming $$p$$ into a corresponding $$n \times n$$ matrix (a companion matrix).
The Abel–Ruffini theorem says that it is impossible to find the roots of an arbitrary polynomial of degree 5 or higher in a finite number of elementary steps (there is no "quintic formula"). That tells us that it is impossible to find the Schur factorization of an arbitrary matrix (exactly) in a finite number of steps for matrices $$5 \times 5$$ or larger. Hence, all Schur algorithms (unlike Hessenberg reductions) are approximate "iterative" algorithms that approach the factor $$T$$ but never exactly reach it (but in practice can quickly approximate $$T$$ to any desired precision).