In Principal Components Analysis (PCA), one gets to the following equation:
$$\psi_l = \frac{1}{\lambda_lN}\sum_{i=1}^n \langle \mathbf{x}_i,\psi_l\rangle\mathbf{x}_i$$
where $(\lambda, \psi)_l$ is an eigenvalue-eigenvector pair, $\mathbf{x}_i$ is the i-th observation of the data and $\langle \cdot \rangle$ denotes the scalar product.
From this equation it is said without any further explanation that $\psi_l$ is a linear combination of the input vectors, i.e.:
$$\psi_l = \sum_{i=1}^n \alpha_{il} \mathbf{x}_i$$
but this is really mysterious: how can it be a linear combination if the right hand has it as a term?