the new cluster variable $x_i'\in\mathfrak{F}$ via the equation $$\label{eq:exchange}
 x_ix_i'=P_i\ .$$

This allows us to refer to the matrix $B$ as the [*exchange matrix*]{} of the cluster $(x_1,\ldots,x_n)$, and to the relations defined by Equation \[eq:exchange\] for $i=1,\ldots,m$ as [*exchange relations*]{}.

We obtain that $(x_1,x_2,\ldots, \hat x_i,x_i',x_{i+1},\ldots, x_n)$ is a transcendence basis of $\mathfrak{F}$. We next construct the new exchange matrix $B_i=B'=(b_{ij}')$, associated to the new (extended) cluster $${\bf x}_i=(x_1,x_2,\ldots, \hat
x_i,x_i',x_{i+1},\ldots, x_n)$$ via its coefficients $b_{ij}'$ as follows:

$\bullet$ $b_{ij}' = -b_{ij}$ if $j \le n$ and $i = k$ or $j = k$,

$\bullet$ $b_{ij}' =  b_{ij} + \frac{|b_{ik} |b_{kj} + b_{ik} |b_{kj} |}{2}$ if $j \le n$ and $i \ne k$ and $j \ne k$,

$\bullet$ $b_{ij}'=b_{ij}$ otherwise.

This algorithm is called [*matrix mutation*]{}. Note that $B_i$ is again skew-symmetrizable (see e.g. [@FZI]). The process of obtaining a new seed is called [*cluster mutation*]{}. The set of seeds obtained from a given seed $({\bf x},B)$ is called the mutation equivalence class of $({\bf x},B)$.

The cluster algebra $\mathfrak{A}\subset \mathfrak{F}$ corresponding to an initial seed $(x_1,\ldots, x_n,B)$ is the subalgebra of $\mathfrak{F}$, generated by the elements of all the clusters in the mutation equivalence class of $({\bf x},B)$ 