Eigenvectors are the arrows a map only stretches
A linear transformation can rotate, shear, or stretch the space around it. Its eigenvectors are the special nonzero directions that refuse to turn—they only scale in place, possibly reversing if the matching eigenvalue happens to be negative.
Apply a linear map T to a nonzero vector v and suppose the output is just λ times v. Then v is an eigenvector and λ its eigenvalue—also called a characteristic value or root. Geometrically the arrow keeps its line; only length changes, or the sense flips when λ is negative. In finite dimensions, choosing a basis turns T into a square matrix A, and the same idea reads Av = λv. Feedback systems amplify the largest |λ| over many iterations, so that eigenvalue governs long-run behavior and its eigenvector the steady direction.
The German prefix eigen means "own" or "characteristic." The concepts began with principal axes of rigid-body rotation and now show up in stability and vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization. A shear of the Mona Lisa that slides upper points right and lower points left leaves horizontal vectors unturned and unscaled—eigenvectors with eigenvalue one. Eigenfunctions arise when the "map" is a differential operator. The full set of eigenvector–eigenvalue pairs is the eigensystem; vectors for one λ plus zero form an eigenspace; an eigenbasis is a basis of eigenvectors.
From the matrix equation one builds the characteristic polynomial det(A − λI), whose roots are the eigenvalues. Diagonalization, when possible, is rewriting A in an eigenbasis so the action becomes pure scaling on each axis. Closely related jargon—eigenbundles, generalized eigenspaces, and more—recycles the same prefix. Wherever a process is linear, asking which directions are merely rescaled is often the fastest way to see the process's skeleton.
Source: Eigenvalues and eigenvectors
