Potential clarification needed - page 229

[gradeyw]

I don't think going from the LHS to the RHS of this equation relies on the upper Hessenberg structure of H (so I don't think "Because H is upper Hessenberg" is necessary)

[EricDarve]

You do need H to be upper Hessenberg. It's not easy to see. f is a polynomial of degree k. We are only interested in the first column of f(H). Take H^2. Since H is upper Hessenberg the first column of H^2 only involves the first 2 columns of H: H^2 = sum_{k=1}^2 H(i,k) H(k,1) Similarly with H^3, we only need the first 3 columns of H. And so on. So if f is a polynomial of degree k f(H) only depends on the first k columns and therefore this is the same as H_k. This is how you can prove that the first k rows, col 1 of f(H) match those of f(H_k).

This does not hold for a general matrix A.