Eigenvalue and Eigenvector Calculator
If analyzing matrices gives you a headache, this eigenvalue and eigenvector calculator is the perfect tool for you. It will allow you to find the eigenvalues of a 2×2 or 3×3 matrix and will even save you time by finding the eigenvectors as well. Enter the matrix elements, click Calculate, and get instant results — including the characteristic polynomial, trace, determinant, all eigenvalues, and their corresponding eigenvectors.
2×2 Matrix
A 2×2 matrix A has the following form:
A = | a₁₁ a₁₂ |
| a₂₁ a₂₂ |
where a₁₁, a₁₂, a₂₁ and a₂₂ are the elements. Our calculator uses exactly this form — make sure to input the numbers in the correct positions.
Calculating the Trace and Determinant
Before finding eigenvalues of a 2×2 matrix, it helps to compute two key numbers:
- Trace — the sum of elements on the main diagonal (top-left to bottom-right):
tr(A) = a₁₁ + a₂₂. It also equals the sum of all eigenvalues. - Determinant — for a 2×2 matrix:
det(A) = a₁₁·a₂₂ − a₁₂·a₂₁. It equals the product of all eigenvalues.
How to Find Eigenvalues (2×2)
Each 2×2 matrix has two eigenvalues λ₁ and λ₂, defined as numbers satisfying A·v = λ·v for a non-zero vector v (the eigenvector). To find them, solve the characteristic equation:
det(A − λI) = 0 λ² − tr(A)·λ + det(A) = 0
Using the quadratic formula:
λ = [ tr(A) ± √(tr(A)² − 4·det(A)) ] / 2
The expression under the square root is the discriminant. If it is negative, the eigenvalues are complex numbers (see below).
Eigenvalue and Eigenvector Calculator — 2×2 Matrices
Once the eigenvalues are known, the eigenvector for each eigenvalue λ is found by solving (A − λI)·v = 0. For a 2×2 matrix this reduces to a single equation:
(a₁₁ − λ)·v₁ + a₁₂·v₂ = 0
Choosing a convenient free variable (e.g. v₂ = 1) gives the eigenvector, which is then normalized to unit length. Our calculator handles this automatically for both eigenvalues.
How to Find Eigenvalues and Eigenvectors of 3×3 Matrices
For a 3×3 matrix, the characteristic polynomial is a cubic:
λ³ − tr(A)·λ² + q·λ − det(A) = 0
where q is the sum of the three 2×2 principal minor determinants. The three roots of this cubic are the eigenvalues. For each real eigenvalue, the corresponding eigenvector is found via Gaussian elimination on the matrix (A − λI). The calculator shows all three eigenvalues and, if they are real, all three eigenvectors.
Complex Eigenvalues and Eigenvectors
When the discriminant of the characteristic polynomial is negative, the matrix has complex eigenvalues, which appear as a conjugate pair:
λ = α ± βi
where α is the real part and β is the imaginary part. Complex eigenvalues occur when a matrix represents a rotation combined with a scaling. The eigenvectors are also complex in this case — the calculator will show the real and imaginary parts of the eigenvalue pair.
Applications
- Physics & Engineering (metric/SI units): vibration analysis, quantum mechanics, moment of inertia tensors, stability of systems of differential equations.
- Engineering (US customary units): structural analysis, mechanical vibrations, principal stresses and strains in materials.
- Finance (USD, EUR, RUB and other currencies): principal component analysis of financial portfolios, covariance matrix decomposition for risk assessment, Markov chain steady-state analysis.
- Computer graphics: 3D transformations, principal axes, image compression (PCA).
- Google PageRank algorithm: the dominant eigenvector of the web link matrix ranks pages.
FAQs
Can a matrix have repeated eigenvalues?
Yes — the characteristic polynomial may have a repeated root (a double or triple eigenvalue). In this case the matrix may or may not have enough linearly independent eigenvectors. The calculator handles repeated roots correctly.
What does it mean if an eigenvalue is zero?
A zero eigenvalue means the matrix is singular (non-invertible) and its determinant is zero. The corresponding eigenvector lies in the null space of the matrix.
Are eigenvectors unique?
No — any non-zero scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue. This calculator returns the normalized (unit-length) eigenvector.
What is the relationship between trace, determinant, and eigenvalues?
For an n×n matrix: tr(A) = λ₁ + λ₂ + … + λₙ and det(A) = λ₁ · λ₂ · … · λₙ. These serve as quick sanity checks for your results.