A Reverse Order Implicit Q-Theorem and the Arnoldi Process

Abstract

Let A be a real square matrix and $V^TAV = G$ be an upper Hessenberg matrix with positive subdiagonal entries, where $V$ is an orthogonal matrix. Then the implicit $Q$-theorem states that once the first column of $V$ is given then $V$ and $G$ are uniquely determined. In this paper, three results are established. First, it holds a reverse order implicit $Q$-theorem: once the last column of $V$ is given, then $V$ and $G$ are uniquely determined too. Second, it is proved that for a Krylov subspace two formulations of the Arnoldi process are equivalent and in one to one correspondence. Finally, by the equivalence relation and the reverse order implicit $Q$-theorem, it is proved that for the Krylov subspace, if the last vector of vector sequence generated by the Arnoldi process is given, then the vector sequence and resulting Hessenberg matrix are uniquely determined.