An eigenvector of a matrix is a special direction that the matrix only stretches or shrinks — never rotates or otherwise distorts. The corresponding eigenvalue is exactly how much it stretches by. These special directions reveal a matrix's essential structure, and finding them ("eigendecomposition") is one of the most important and widely applied computational techniques in all of applied mathematics.
Picture a matrix as a machine that stretches, squishes, and rotates space (see the Matrix entry). For most directions you could point in, the machine will do all three at once — some stretching, some rotation. But for certain very special directions, something remarkable happens: the machine only stretches or shrinks along that exact direction — it doesn't rotate it at all. These special directions are called eigenvectors, and the amount of stretching is the corresponding eigenvalue.
A non-zero vector v is an eigenvector of matrix A, with eigenvalue λ, if multiplying v by A gives back exactly λ times v — the same direction, just scaled. This single equation captures the entire "doesn't rotate, only stretches" intuition precisely.
Rearranging Av=λv gives (A−λI)v=0. For this to have a non-zero solution v, the matrix (A−λI) must be non-invertible, meaning det(A−λI)=0 (see the Determinant entry). This equation, expanded out, is a polynomial in λ called the characteristic polynomial — its roots are exactly the eigenvalues of A.
For A = [[2,0],[0,3]] (a diagonal matrix), the eigenvalues are simply the diagonal entries themselves, 2 and 3, with eigenvectors (1,0) and (0,1) respectively — diagonal matrices make their eigenstructure immediately visible.
If an n×n matrix has n linearly independent eigenvectors, it can be diagonalised: written as A = PDP⁻¹, where D is a diagonal matrix of eigenvalues and P has the eigenvectors as its columns. This is enormously useful computationally: raising A to a high power (Aⁿ) becomes easy, since Aⁿ = PDⁿP⁻¹, and Dⁿ is trivial to compute (just raise each diagonal eigenvalue to the n-th power).
Eigenvalues can be complex numbers even for real-valued matrices (typically corresponding to rotational behaviour in the transformation), and a matrix can have repeated eigenvalues that don't always come with enough independent eigenvectors to allow full diagonalisation — in such "defective" cases, the more general Jordan normal form provides the closest achievable analogue to diagonalisation.
For real symmetric matrices (A=Aᵀ), a particularly beautiful result holds: all eigenvalues are guaranteed to be real (never complex), and the matrix always has a full set of eigenvectors that are mutually orthogonal (perpendicular) to each other — meaning symmetric matrices are always fully diagonalisable using an orthogonal basis. This spectral theorem is one of the most useful results in linear algebra, and its complex-number generalisation (for Hermitian matrices) is precisely what makes quantum mechanical observables mathematically well-behaved.
Google's original PageRank algorithm (Brin and Page, 1998) models the entire web as a matrix where each entry represents the probability of randomly clicking from one page to another. The algorithm's key insight: a page's "importance" score should correspond to the steady-state probability of a random web-surfer being on that page after clicking links indefinitely — mathematically, this steady-state distribution is exactly the eigenvector corresponding to the largest eigenvalue (which, for this specific type of matrix, is always exactly 1) of the web's link matrix. Computing this dominant eigenvector for a matrix representing billions of web pages requires specialised, highly efficient numerical algorithms (the power iteration method being the simplest conceptual approach).
In quantum mechanics, physical observables (position, momentum, energy, spin) are represented mathematically as Hermitian operators (the complex-number analogue of symmetric matrices) acting on a Hilbert space (see the Vector Space entry). The only possible values you could ever actually measure for that observable are exactly its eigenvalues, and after a measurement, the quantum system's state "collapses" into the corresponding eigenvector. This isn't merely a convenient mathematical analogy — eigenvalues and eigenvectors are the literal, foundational mathematical language in which the postulates of quantum mechanics are formally stated.
For a matrix with all non-negative entries (like PageRank's web-link matrix, or a Markov chain's transition matrix), the Perron-Frobenius theorem (developed by Oskar Perron, 1907, and Georg Frobenius, 1912) guarantees the existence of a uniquely largest, real, positive eigenvalue with a corresponding eigenvector that has all non-negative entries — a foundational and widely applied result underlying the mathematical guarantee that PageRank-style algorithms and Markov chain long-run behaviour analyses are always well-defined and computationally tractable.