Eigenvalue & Eigenvector Calculator

What Are Eigenvalues and Eigenvectors?

An eigenvector of a square matrix A is a non-zero vector v that, when multiplied by A, yields a scalar multiple of itself. That scalar is called the eigenvalue. In equation form:

The word "eigen" comes from the German word meaning "own" or "characteristic." Eigenvalues and eigenvectors reveal the intrinsic properties of a linear transformation: the directions along which the transformation acts as simple scaling, and the scale factors themselves.

Every n×n matrix has exactly n eigenvalues (counted with multiplicity) over the complex numbers, a fundamental result guaranteed by the Fundamental Theorem of Algebra. Real matrices may have complex eigenvalues, which always appear in conjugate pairs.

The Eigendecomposition (Spectral Decomposition)

If a square matrix A has n linearly independent eigenvectors, it can be factored as:

where V is a matrix whose columns are the eigenvectors of A, and D is a diagonal matrix containing the corresponding eigenvalues along its diagonal. This factorization is called the eigendecomposition or spectral decomposition.

Not every matrix can be eigendecomposed. A matrix is diagonalizable if and only if it has a full set of n linearly independent eigenvectors. Defective matrices (those lacking a complete set of independent eigenvectors) cannot be diagonalized, though they can be reduced to Jordan normal form.

Geometric Interpretation

Geometrically, multiplying by a matrix represents a linear transformation: rotations, reflections, scalings, and shears. Eigenvectors are the special directions that remain unchanged (or simply reverse) under the transformation. The eigenvalue tells you the stretching factor along that direction:

• λ > 1 — the eigenvector direction is stretched
• 0 < λ < 1 — the eigenvector direction is compressed
• λ = 1 — the direction is unchanged (fixed point of the transformation)
• λ = 0 — the direction collapses to zero (the matrix is singular)
• λ < 0 — the direction is reversed and scaled by |λ|
• λ is complex — the transformation involves rotation in a 2D subspace

Think of a matrix as a machine that distorts space. The eigenvectors point along the axes of that distortion, and the eigenvalues measure how much distortion happens along each axis.

The Characteristic Polynomial

Eigenvalues are the roots of the characteristic polynomial, obtained by solving:

For an n×n matrix, this produces a degree-n polynomial in λ. For a 2×2 matrix [[a, b], [c, d]], the characteristic polynomial is:

where (a + d) is the trace and (ad - bc) is the determinant. For larger matrices, directly solving the characteristic polynomial is numerically unstable, which is why iterative algorithms like QR iteration are used in practice.

The QR Algorithm

The QR algorithm is the standard numerical method for computing eigenvalues. It works by repeatedly factoring a matrix into an orthogonal matrix Q and an upper triangular matrix R, then multiplying them in reverse order:

1. Initialize: Set A₀ = A
2. Iterate: For k = 0, 1, 2, ..., compute the QR factorization A_k = Q_kR_k
3. Update: Form A_k+1 = R_kQ_k
4. Converge: A_k converges to an upper triangular (or quasi-triangular) matrix whose diagonal entries are the eigenvalues

The QR algorithm preserves eigenvalues at each step because A_k+1 = Q_k^TA_kQ_k is a similarity transformation. In practice, the algorithm is accelerated using Wilkinson shifts and an initial Hessenberg reduction, bringing the complexity from O(n⁴) down to roughly O(n³) per run.

Special Cases: Symmetric Matrices

Real symmetric matrices (where A = A^T) enjoy especially nice eigendecomposition properties guaranteed by the Spectral Theorem:

• All eigenvalues are real — no complex eigenvalues can appear
• Eigenvectors are orthogonal — eigenvectors corresponding to distinct eigenvalues are automatically perpendicular
• Always diagonalizable — even with repeated eigenvalues, a full set of orthogonal eigenvectors exists
• Decomposition simplifies: A = QDQ^T where Q is orthogonal (Q^-1 = Q^T)

The symmetric eigendecomposition is the foundation of Principal Component Analysis (PCA), where the covariance matrix (always symmetric positive semi-definite) is decomposed to find the directions of greatest variance in a dataset.

Applications of Eigendecomposition

Eigenvalues and eigenvectors appear across virtually every area of science and engineering:

• Principal Component Analysis (PCA) — eigenvectors of the covariance matrix identify the principal axes of variation in data, enabling dimensionality reduction and feature extraction
• Google PageRank — the dominant eigenvector of the web's link matrix determines the relative importance of every webpage on the internet
• Quantum Mechanics — observable quantities correspond to eigenvalues of Hermitian operators; a measurement always yields an eigenvalue, and the system collapses to the corresponding eigenvector (eigenstate)
• Vibration Analysis — eigenvalues of the stiffness-mass matrix system determine the natural frequencies of a structure, while eigenvectors describe the mode shapes
• Stability Analysis — the eigenvalues of the Jacobian matrix of a dynamical system determine stability: all eigenvalues with negative real parts mean the equilibrium is stable
• Markov Chains — the stationary distribution of a Markov chain is the eigenvector corresponding to eigenvalue 1 of the transition matrix
• Differential Equations — systems of linear ODEs dx/dt = Ax are solved via eigendecomposition, yielding solutions x(t) = Ve^DtV^-1x(0) with exponentials of the eigenvalues
• Image & Signal Processing — eigenfaces for facial recognition, Karhunen-Loève transform for signal compression
• Graph Theory — eigenvalues of the adjacency matrix and Laplacian matrix reveal graph connectivity, clustering structure, and expansion properties
• Control Theory — eigenvalues of the state-transition matrix determine system poles, response speed, and controllability

Computational Complexity

Computing eigenvalues and eigenvectors is significantly more expensive than many other matrix operations:

• Characteristic polynomial: Finding roots of the degree-n polynomial costs O(n³) to form the polynomial, but root-finding is numerically unstable for large n
• QR algorithm: O(n³) per iteration, with typically O(n) iterations needed for convergence, giving an overall complexity of roughly O(n³) after the initial Hessenberg reduction
• Symmetric case: Specialized algorithms (tridiagonal QR, divide-and-conquer, MRRR) bring the cost to O(n²) for eigenvalues only and O(n³) with eigenvectors
• Sparse matrices: Iterative methods like Lanczos (symmetric) and Arnoldi (general) can find a few eigenvalues in O(n) time per iteration, making them practical for large-scale problems

For the matrices supported by this calculator (up to 5×5), computation is instantaneous. Industrial applications use optimized LAPACK routines for matrices with thousands or millions of rows.

Connection to Other Decompositions

Eigendecomposition is deeply connected to the other matrix decompositions available on this site:

• SVD (Singular Value Decomposition) — the singular values of A are the square roots of the eigenvalues of A^TA. For symmetric positive semi-definite matrices, SVD and eigendecomposition coincide. SVD applies to any matrix (including rectangular), while eigendecomposition requires square matrices
• QR Decomposition — the QR algorithm for eigenvalues is built on repeated QR factorizations. QR decomposition is a single factorization step; eigendecomposition is the converged result of iterating it
• LU Decomposition — while LU solves linear systems directly, eigendecomposition transforms the system into a diagonal one. The determinant of A equals the product of its eigenvalues, connecting to the pivots found during LU factorization
• Cholesky Decomposition — a symmetric positive definite matrix (all eigenvalues positive) can be Cholesky-factored as A = LL^T. The Cholesky factor L relates to the eigenvectors scaled by the square roots of the eigenvalues

Key Properties of Eigenvalues

• The trace of a matrix equals the sum of its eigenvalues: tr(A) = λ₁ + λ₂ + ... + λ_n
• The determinant equals the product of the eigenvalues: det(A) = λ₁ · λ₂ · ... · λ_n
• A matrix is singular if and only if at least one eigenvalue is zero
• A matrix is positive definite if and only if all eigenvalues are strictly positive
• The eigenvalues of A^k are λ_i^k (the eigenvalues raised to the k-th power)
• The eigenvalues of A^-1 are 1/λ_i (the reciprocals of the eigenvalues)
• Similar matrices (B = P^-1AP) have identical eigenvalues
• The spectral radius ρ(A) = max|λ_i| governs the convergence behavior of iterative methods