of the subsection. We introduce a $(s+1)$-component vector $\mathbf{A}(z)$ depending on a spectral parameter and with algebraic-valued (*i.e* non-commutative) entries. These algebraic elements will be basically the matrices entering the matrix product construction of the stationary state.

The construction relies essentially on two key relations. The Zamolodchilov-Faddeev (ZF) relation encodes the commutation relations of the matrix ansatz algebra $$\label{eq:ZF}
 \check{R}\left(\frac{z_1}{z_2}\right) \mathbf{A}(z_1) \otimes \mathbf{A}(z_2) = \mathbf{A}(z_2) \otimes \mathbf{A}(z_1).$$ The associativity of the algebra is ensured by the Yang-Baxter equation . Another consistency relation is ensured by the unitarity property .

The Ghoshal-Zamolodchilov (GZ) relations encode the action of the matrix ansatz algebra on the boundary vectors ${\langle\!\langle}W|$ and $|V{\rangle\!\rangle}$ (which are used to contract the matrix product into a scalar) $$\label{eq:GZ}
  {\langle\!\langle}W| \left( K(z)\mathbf{A}\left(\frac{1}{z}\right)-\mathbf{A}(z) \right)=0, \qquad 
  \left(\overline{K}(z)\mathbf{A}\left(\frac{1}{z}\right)-\mathbf{A}(z) \right)|V{\rangle\!\rangle}= 0.$$ The consistency between these actions and the commutation relations of the matrix ansatz algebra (*i.e* the ZF relation) is ensured by the reflection equations and .

We can now introduce the matrix product state $$\label{eq:inhomogeneous_ground_state}
 {{|\cS(z_1,z_2,\dots,z_L)\rangle}}=\frac{1}{Z_L(z_1,z_2,\dots,z_L)}
 {\langle\!\langle}W| \mathbf{A}(z_1) \otimes \mathbf{A}(z_2) \otimes \dots \otimes \mathbf{A}(z_L) |V{\rangle\!\rangle},$$ with $Z_L(z_1,z_2,\dots,z_L) = {\langle\!\langle}W|\bm{C}(z_1)\bm{C}(z_2) \cdots \bm{C}(z_L)|V{\rangle\!\rangle}$ where the algebraic element $\bm{C}(z)$ is defined as the sum of the components of the vector $\bm{A}(z)$ $$\bm{C}(z)={{\langle\sigma|}}\bm{A}(z).$$ It has been 