An eigenvector of matrix A is a non-zero vector v where Av = λv - multiplying by A only scales v, not rotates it. λ is the eigenvalue, the scaling factor.

Eigenvectors reveal the "natural axes" of a transformation - directions that stay fixed. Eigenvalues tell you how much stretching happens along those axes. In PCA, the eigenvectors of the covariance matrix are the principal components (directions of maximum variance), and eigenvalues indicate how much variance each explains. In graph neural networks and spectral methods, eigenvalues of the graph Laplacian encode structural properties of the graph.