Eigenvalues 2×2 Matrix

Eigenvalues of a 2×2 matrix: characteristic equation

Take a matrix A = [[a, b], [c, d]]. Its eigenvalues λ are the scalars for which A·v = λ·v holds with some non-zero eigenvector v. Those values turn out to be the roots of the characteristic equation det(A − λI) = 0. For a 2×2 matrix that expands neatly into λ² − tr(A)·λ + det(A) = 0, where the trace is tr(A) = a + d and the determinant is det(A) = ad − bc. Solving it gives the closed form λ = (tr ± √(tr² − 4·det)) / 2. What the discriminant Δ = tr² − 4·det tells you is the kind of answer you'll get. With Δ > 0 you have two distinct real eigenvalues; Δ = 0 means a single repeated real one; Δ < 0 gives a complex conjugate pair. And by the spectral theorem, any symmetric matrix is guaranteed real eigenvalues and orthogonal eigenvectors.

Applications: PCA, stability and quantum mechanics

You run into eigenvalues all over applied math. In machine learning they power Principal Component Analysis (PCA), where the largest eigenvalue of the covariance matrix marks the direction of maximum variance. In dynamical systems they drive stability analysis, since the signs of the Jacobian's eigenvalues classify each equilibrium. Engineers use them for structural vibration modes; Markov chains rely on them too, with PageRank built on the principal eigenvector; in quantum mechanics the eigenvalues of a Hermitian operator are the observables; and they show up again in spectral tomography.

FAQ

Can a real matrix have complex eigenvalues? It can. Rotation matrices are the classic example, with a conjugate pair sitting on the unit circle.

What does a zero eigenvalue mean? That the matrix is singular and can't be inverted, since det(A) = 0.

Are eigenvalues unique? The set of them is. Eigenvectors, on the other hand, are only fixed up to scaling.

How to find eigenvectors? For each eigenvalue λ, solve (A − λI)·v = 0.