grade, this choice is not very critical and we will typically reach the same fixed point with other initial seeds. Similarly, the choice of the exact value of $\alpha$ is not critical and largely affects the speed of convergence. This is because the fixed point is an eigenvector of the grade matrix $A$.

The PeerRank rule returns grades that are an eigenvector of the grade matrix $A$.

[[**Proof:  **]{}\
]{}In matrix notation, at the fixed point, we have: $$X = (1-\alpha)X + \frac{\alpha}{|X|} A.X$$ That is, $$X = X - \alpha X + \frac{\alpha}{|X|} A.X$$ Rearranging and cancelling gives: $$\frac{\alpha}{|X|} A.X = \alpha X$$ Dividing by $\alpha$ and letting $\lambda = |X|$, we get: $$A.X = \lambda X$$

SOME EXAMPLES
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To illustrate how the PeerRank rule works on some simple cases, we consider a few examples.

Unanimous grade matrix {#unanimous-grade-matrix .unnumbered}
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Suppose that every entry in the grade matrix $A$ is the grade $k$ with $0 \leq k \leq 1$. Now an eigenvector of $A$, and the PeerRank solution assigns each agent with this grade $k$. The weighted average of identical grades is always the same whatever the weights. This is what we might expect. The grade matrix 