I learn kernel PCA from wikipedia. In this article, the eigen equation is
\begin{equation} N \lambda \vec{\alpha} = \boldsymbol{K} \vec{\alpha} \end{equation} where $\lambda$ is the eigen value, $\vec{\alpha}$ is the eigen vector of $\lambda$ and $\boldsymbol{K}$ is the kernel matrix.
Why is the left side multiplied by N? What is the meaning and effect of $N$?
Recall for linear PCA $N$ is the number of data points in the set and comes in the eigen decomposition of the covariance matrix $C$, where $C = \frac{1}{N} \sum_{i=1}^N x_i x_i^\top$ ( we look for eigenvectors $v$ such that $\lambda v = C v$). So we simply multiply through by the constant $N$ to bring it through to the other side. It is similar multiplication by the constant when using a kernel method.