The Arnoldi decomposition states the following: let $A \in \mathbb{C}^{n×n}$ and $v ∈ \mathbb{C}^{n} \setminus \{0\}$ be of grade $d$ with respect to $A$. Then there exists $V \in \mathbb{C}^{n×d}$ with orthonormal columns and an unreduced upper Hessenberg matrix $H \in \mathbb{C}^{d×d}$, s.t. $$ AV = V H $$
The columns of $V$ form an orthonormal basis of the $d$-th Krylov subspace of $A$ and $v$ which is defined as
$$ \mathcal{K}_d(A,V) := \operatorname{span}\{v,Av, \dots, A^{d-1}v\} $$
That is why we can apply Gram-Schmidt to the matrix $W:= [v,Av, \dots, A^{d-1}v]$ in order to compute the Arnoldi decomposition. I was wondering if this decomposition is in fact unique. I would assume yes, because in the proof we use that
$$W = [V, \tilde{V}] \begin{bmatrix} R \\ 0 \end{bmatrix} $$
with $V \in \mathbb{C}^{n×d}$ and $R \in \mathbb{C}^{d×d}$ being the unique QR decomposition as $W$ has full rank.
But also when you derive the Arnoldi algorithm, you will see that $v_1 = \frac{v}{\|v\|}$ and $Av_j = h_{j+1,j}v_{j+1} + \sum_{i=1}^{j}h_{i,j}v_i$. If you then use the "orthonormality constraint" you see exactly that $h_{i,j} = \langle Av_j, v_i \rangle$ and $h_{j+1,j} = \|\hat{v}_{j+1}\|$ where $\hat{v}_{j+1} = A{v_j} − \sum_{i=1}^{j}h_{i,j}v_i$, which is also a unique solution "structure".
So to reiterate my question: Are $V,H$ unique in the Arnoldi decomposition? If not then how does this correspond to the Arnoldi algorithm?