Eigenvectors & Eigenvalues

The defining equation

A matrix A normally pushes a vector off its original line — it rotates it, stretches it, or both. An eigenvector of A is a special nonzero vector v that A only stretches or shrinks, never rotates off its own line: Av = lambda v, where the scalar lambda is the matching eigenvalue. Geometrically, eigenvectors are the invariant axes of the linear transformation — a whole coordinate system that the matrix treats as pure scaling.

A v = lambda v
(A - lambda I) v = 0,   v != 0   requires   det(A - lambda I) = 0

The characteristic polynomial

For a 2x2 matrix, expanding det(A - lambda I) = 0 gives a quadratic in lambda whose coefficients are just the matrix's trace (sum of the diagonal) and determinant. Solving that quadratic hands you both eigenvalues directly, and the discriminant tells you what kind of eigenvalues you're dealing with before you even finish the arithmetic.

for A = [[a, b], [c, d]]:

lambda^2 - (a+d) lambda + (a*d - b*c) = 0

lambda = [ (a+d) +/- sqrt( (a+d)^2 - 4(a*d - b*c) ) ] / 2

live demo · the unit circle deforming into an ellipse aligned with the eigenvectors● LIVE

Complex and repeated eigenvalues: rotation and shear

When the discriminant is negative, the two eigenvalues are complex conjugates, which is the algebra's way of telling you that the matrix has no real invariant direction at all — it is a rotation combined with scaling, and a pure rotation matrix has eigenvalues e^(+/-i*theta) sitting on the unit circle in the complex plane. A repeated real eigenvalue is a different edge case: if the matrix happens to be diagonalizable it acts as uniform scaling in every direction (the ellipse becomes a circle again), but if it is "defective" it behaves as a shear that needs a generalized eigenvector to fully describe, since a single eigenvector direction cannot span the whole plane's worth of behaviour.

Why the unit circle becomes an ellipse aligned with the eigenvectors

For a symmetric matrix, the spectral theorem guarantees real eigenvalues and mutually orthogonal eigenvectors. Apply that matrix to every point on the unit circle and each point is stretched by a factor equal to whichever eigenvalue governs the direction it sits closest to, so the circle deforms into an ellipse whose axes point exactly along the eigenvectors, with semi-axis lengths equal to the absolute values of the eigenvalues. This is the geometric heart of the singular value decomposition and of PCA (see this site's PCA visualiser): the principal axes of a transformed shape reveal the matrix's own natural coordinate system.

Where this shows up: PCA, stability, vibration modes

Principal Component Analysis diagonalizes a dataset's covariance matrix; its eigenvectors are the directions of maximum variance and its eigenvalues are the variance explained along each one. Stability analysis of a dynamical system's fixed points (see this site's phase-portrait article) reads the sign and type of the Jacobian's eigenvalues to classify a fixed point as a node, spiral, saddle or center. In structural engineering, the natural vibration modes and resonant frequencies of a bridge or building are literally the eigenvectors and eigenvalues of its stiffness and mass matrices — three very different fields, the same underlying algebra.

Frequently asked questions

What does it mean for a matrix to have complex eigenvalues?

It means there is no real direction that the matrix simply stretches without rotating. Complex eigenvalues always come in conjugate pairs for a real matrix, and geometrically they signal that the transformation includes a rotation component, such as a pure rotation matrix, whose eigenvalues sit on the unit circle in the complex plane at angle equal to the rotation angle.

Why does the unit circle turn into an ellipse aligned with the eigenvectors?

For a symmetric matrix, the spectral theorem guarantees real, mutually orthogonal eigenvectors. Applying the matrix to every point on the unit circle stretches the circle by a factor equal to each eigenvalue exactly along its own eigenvector direction, and because those directions are perpendicular, the result is an ellipse whose axes point along the eigenvectors and whose semi-axis lengths equal the absolute values of the eigenvalues.

How are eigenvectors used in Principal Component Analysis?

PCA diagonalises the covariance matrix of a dataset. Its eigenvectors are the principal component directions — the axes along which the data varies the most — and its eigenvalues equal the variance explained along each of those directions, so ranking eigenvalues from largest to smallest ranks the components by how much of the dataset's spread they capture.

Try it live

Everything above runs in your browser — open Eigenvectors & Eigenvalues and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Eigenvectors & Eigenvalues simulation

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