Eigenvalue Calculator
Calculate eigenvalues and eigenvectors of 2x2, 3x3, and 4x4 matrices. Find the characteristic polynomial, multiplicities, and diagonalizability.
Quick Examples
Enter values for each cell of the matrix
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Eigenvalue and Eigenvector Calculator
Calculate eigenvalues and eigenvectors for any 2×2, 3×3, or 4×4 matrix instantly. Our calculator shows the characteristic polynomial, computes all eigenvalues (including complex numbers), finds corresponding eigenvectors, and determines if your matrix is diagonalizable—all with complete step-by-step solutions.
What Are Eigenvalues and Eigenvectors?
Eigenvalues (λ) and eigenvectors (v) are fundamental concepts in linear algebra. For a square matrix A, an eigenvector is a non-zero vector that, when multiplied by A, only gets scaled by a factor (the eigenvalue): Av = λv. Eigenvalues are found by solving the characteristic equation det(A - λI) = 0. They reveal the matrix's intrinsic properties: how it stretches, rotates, or reflects vectors.
Characteristic Equation
det(A - λI) = 0How to Calculate Eigenvalues
Select the matrix size: 2×2, 3×3, or 4×4
Enter the matrix elements in the input grid (row by row)
Or choose a preset example to see how different matrix types behave
View the characteristic polynomial p(λ) and all eigenvalues
See the corresponding eigenvector for each eigenvalue
Check algebraic and geometric multiplicities to understand eigenspace dimensions
Eigenvalue Applications
Principal Component Analysis (PCA)
Find the covariance matrix eigenvalues to identify the most important directions of variation in your data.
Vibration Analysis
Eigenvalues determine natural frequencies of mechanical systems—critical for bridge and building design.
Quantum Mechanics
Observable quantities in quantum physics are eigenvalues of Hermitian operators representing physical measurements.
Stability Analysis
Eigenvalues of a system's Jacobian matrix determine if equilibrium points are stable, unstable, or oscillatory.
Google PageRank
The PageRank algorithm uses the principal eigenvector of the web link matrix to rank webpage importance.
Image Compression
Eigenvalue decomposition enables image compression by keeping only the largest eigenvalues and their eigenvectors.
Why Use This Calculator
Avoid Calculation Errors
Finding eigenvalues requires solving polynomial equations and careful determinant calculations. Our calculator ensures accurate results every time.
Complex Eigenvalue Support
Handles matrices with complex eigenvalues (like rotation matrices) that are difficult to compute by hand.
Step-by-Step Solutions
See the complete derivation: characteristic polynomial, eigenvalue solving, and eigenvector calculation.
Multiple Matrix Sizes
Work with 2×2, 3×3, or 4×4 matrices—automatically adapts to your needs.
Complete Analysis
Get algebraic and geometric multiplicities, plus diagonalizability status for full matrix characterization.
Frequently Asked Questions
For a 2×2 matrix [[a,b],[c,d]], solve the characteristic equation det(A - λI) = λ² - (a+d)λ + (ad-bc) = 0. Use the quadratic formula: λ = [(a+d) ± √((a+d)² - 4(ad-bc))] / 2. The trace (a+d) equals the sum of eigenvalues, and the determinant (ad-bc) equals their product.
Complex eigenvalues occur when the characteristic polynomial has no real roots—typically in rotation matrices. For example, a 90° rotation matrix has eigenvalues λ = ±i. Complex eigenvalues always come in conjugate pairs (a+bi and a-bi) for real matrices.
The characteristic polynomial is det(A - λI), where I is the identity matrix. For an n×n matrix, it's a polynomial of degree n in λ. Its roots are the eigenvalues. For a 2×2 matrix: λ² - trace(A)·λ + det(A).
Algebraic multiplicity is how many times an eigenvalue appears as a root of the characteristic polynomial. Geometric multiplicity is the dimension of its eigenspace (number of linearly independent eigenvectors). A matrix is diagonalizable if and only if algebraic = geometric multiplicity for all eigenvalues.
A matrix is diagonalizable if it has n linearly independent eigenvectors (for an n×n matrix). This happens when geometric multiplicity equals algebraic multiplicity for each eigenvalue. Symmetric matrices are always diagonalizable. Defective matrices (like shear matrices) are not diagonalizable.
For any square matrix: Trace = sum of all eigenvalues, Determinant = product of all eigenvalues. For a 2×2 with eigenvalues λ₁ and λ₂: trace = λ₁ + λ₂ and det = λ₁ × λ₂. This provides a quick way to verify your eigenvalue calculations.