Eigenvalues & Eigenvectors: The Directions a Matrix Doesn't Rotate

The vectors a matrix doesn't rotate

A matrix A acting on a vector generally both rotates and scales it. An eigenvector is one of the special directions where A does only the scaling — the output vector points along the exact same line as the input, just longer, shorter, or flipped. The scale factor is the eigenvalue λ:

A v = λ v          (v ≠ 0)

Visually, apply A to every point on the unit circle and the image is generally an ellipse. The eigenvectors are exactly the two directions where a point on the circle maps to a point on the ellipse along the same line from the origin — everywhere else on the circle, the image point has rotated off that radial line. The eigenvalues are how much the ellipse stretches along those two special axes; the major and minor axis lengths of the image ellipse are literally |λ1| and |λ2| when the eigenvectors are orthogonal.

Finding them: the characteristic equation

Rearranging Av = λv gives (A − λI)v = 0. For a nonzero v to solve this, the matrix (A − λI) must be singular — non-invertible — which happens exactly when its determinant is zero. That determinant, expanded, is a polynomial in λ called the characteristic polynomial, and its roots are the eigenvalues:

det(A − λI) = 0
for A = [[a,b],[c,d]]:
  λ² − (a+d)λ + (ad−bc) = 0
  λ² − trace(A)·λ + det(A) = 0

For a 2×2 matrix this is just a quadratic, solvable with the quadratic formula, and its two coefficients are the trace (sum of diagonal entries, equal to the sum of the eigenvalues) and the determinant (equal to the product of the eigenvalues) — a useful sanity check when computing eigenvalues by hand.

Real, repeated, or complex — and what each looks like

The discriminant of that quadratic, trace² − 4·det, decides the character of the eigenvalues. Positive: two distinct real eigenvalues, two genuine fixed directions, the ellipse has two different-length axes. Zero: a repeated real eigenvalue — either the matrix scales every direction equally (a pure scaling, e.g. the identity times a constant) or, more commonly for a non-diagonal matrix, there's only one independent eigenvector direction and the transformation includes a shear alongside the scaling. Negative: a pair of complex conjugate eigenvalues, which means there is no real direction left unrotated — geometrically, the transformation is dominated by rotation (mixed with scaling by the eigenvalues' magnitude), and repeatedly applying A traces a vector spiralling around the origin rather than converging along any fixed line.

Why this matters far beyond geometry

Eigenvalues describe the long-run behaviour of any system that evolves by repeatedly applying the same linear transformation. In a discrete dynamical system xₙ₊₁ = Axₙ, the eigenvalue with the largest magnitude dominates as n grows — if that magnitude exceeds 1 the system blows up along that eigenvector's direction, if it's below 1 the system decays toward the origin along it, and this single fact underlies stability analysis for everything from population models to control systems to the convergence of iterative numerical algorithms. In statistics, the eigenvectors of a covariance matrix are the principal components — the directions of maximum variance in a dataset — and their eigenvalues are exactly how much variance lies along each one, which is the entire mechanism behind PCA dimensionality reduction.

Diagonalization: why eigenvectors are the 'natural' basis

If a matrix has enough independent eigenvectors to form a full basis, you can write A = PDP⁻¹, where D is diagonal (the eigenvalues on the diagonal) and P's columns are the eigenvectors. In that eigenvector basis, A's action is nothing but independent scaling along each axis — all the geometric complexity of a general matrix reduces to a list of numbers. This is also why computing Aⁿ for large n is cheap once you have the decomposition: Aⁿ = PDⁿP⁻¹, and raising a diagonal matrix to a power just means raising each eigenvalue to that power, instead of n expensive matrix multiplications.

Frequently asked questions

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