Let $p_n$ be a monic polynomial of degree n (a polynomial of degree n with the coefficient of the term of degree n equal to 1).
Let A be a square matrix and $A=Q^*HQ$ where H is a Hessenberg matrix. Given a vector b, why can the polynomial $p(A)b$ be written as $A^nb - Qy$ for some vector y?
This is the first step in proving the following property of the arnoldi iteration, which I am trying to understand.
"The matrix Hn can be characterized by the following optimality condition. The characteristic polynomial of Hn minimizes ||p(A)q1||2 among all monic polynomials of degree n. This optimality problem has a unique solution if and only if the Arnoldi iteration does not break down."
The question should be edited in this way: any monic polynomial p(A) can be written as $A^n-Qy$ where y is an arbitrary vector and Q is the orthonormal basis for $K_n$ (The $n_{th}$ Krylov subspace of A). Hence, the answer to this question becomes straightforward: The term with the highest degree of the monic polynomial would be by definition $A^n$ and the rest of polynomial can be formed by Q which is the orthonormal basis for $K_n:=<b,Ab,A^2b,...,A^{n-1}b>$ For a more comprehensive elaboration on Arnoldi iteration and Krylov subspaces you can look into numerical linear algebra handbooks e.g. Chapter 34 "Numerical Linear Algebra" by Lloyd N. Trefethen.