Normalizing the Eigenvectors

Next we choose the normalization of these eigenvectors so that the largest element (or one of the equally largest elements) has value 1. Let's call this element $k$, and we can therefore bound the magnitude of the other elements to be less than or equal to 1. This leads to the inequalities, using the property that $M_{ij}\geq 0$,

$$\begin{eqnarray} \sum_i M_{ik} \leq \lambda_{\rm max} \nonumber\\ M_{kk}-\sum_{i \neq k} M_{ik} \geq \lambda_{\rm min} \end{eqnarray}$$

where the equality from the maximum will occur only if the eigenvector takes the value 1 for all values of $i$ where $M_{ik} \neq 0$, and the equality for the minimum will occur only if the eigenvector takes the value -1 for all values of $i\neq k$ where $M_{ik} \neq 0$.

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