At each iteration i of the GMRES method, is calculated a single new orthonormal vector of the existing Krylov subspace. If the norm of that vector is 0 (or close to 0), then the subspace is "complete", and we should be able to calculate the converged solution. In theory, this occurs when we have N orthonormal vectors (where N is the size of A).

The solution can then be computed using x=x0+Vy, where V is the matrix of orthonormal vectors computed above. [x0 (Nx1),V(NxN) and y(Nx1)].

My question is: What happens if the algorithm find only P < N orthonormal vectors. How can I complete the Krylov-subspace and how can be computed x=x0+Vy if V(NxP) is not a square matrix?

At each iteration $i$ of the GMRES method, is calculated a single new orthonormal vector of the existing Krylov subspace. If the norm of that vector is 0 (or close to 0), then the subspace is "complete", and we should be able to calculate the converged solution. In theory, this occurs when we have $N$ orthonormal vectors (where $N$ is the size of $A$).

The solution can then be computed using $$x=x_0+Vy$$ where $V$ is the matrix of orthonormal vectors computed above. [$x_0 (N \times 1),V(N \times N)$ and $y(N \times 1)$].

My question is: What happens if the algorithm finds only $P < N$ orthonormal vectors? How can I complete the Krylov-subspace and how can be computed $x=x_0+Vy$ if $V(N \times P)$ is not a square matrix?