Eigenvalues & Eigenvectors

Stretch a sheet of rubber in some direction. Most lines on it get skewed — but one line through the stretch stays straight, just longer. That's an eigenvector: a direction the transformation leaves unrotated, only scaled. The scaling factor is the eigenvalue. Eigenvectors reveal a matrix's natural axes — and they power Google's PageRank, PCA in data science, and every vibration analysis in engineering.