where $B = F_{n,k-1}$ for MPE, $B = F_{n+1,k}- F_{n,k-1}= F_{n,k}\Delta$ for RRE, and $B =\begin{bmatrix}\vec{q}_1&\dots&\vec{q}_k \end{bmatrix}$ for MMPE.
TEA cannot be expressed in this form in general.
If the columns of $F_{n,k}$ form a Krylov subspace, that is $\fxi{n+1}= A \fxi{n}$ for all $n$, then these methods are Krylov methods.
For example, MPE now requires that $F_{n,k}\vec{u}$ is orthogonal to all $\fxi{n+i}$ for $i < k$.
Therefore, the residual $F_{n,k}\vec{u}$ is orthogonal to the Krylov subspace $\mathcal{K}_{k-1}(A,\fxi{n})$.
This is equivalent then to GMRES (see Sidi and Walter \& Ni for details).
TEA may now be expressed as the form above with $B =\begin{bmatrix}\vec{q}& A^\top\vec{q}&\dots&(A^\top)^{k-1}\vec{q}\end{bmatrix}$.
Table \ref{tab:KrylovExtrap} gives the orthogonalization conditions and corresponding Krylov methods for these four extrapolation methods.