liminfoEigenvalue Calculator
Free web tool: Eigenvalue Calculator
Matrix
Eigenvalues
Steps:
Characteristic equation: det(A - λI) = 0
\u03BB\u00B2 - 7\u03BB + 10 = 0
Discriminant: 9.0000
About Eigenvalue Calculator
The Eigenvalue Calculator is a free browser-based linear algebra tool that computes eigenvalues and eigenvectors for square matrices of size 2×2 and 3×3. For a 2×2 matrix, the tool uses the closed-form characteristic equation: λ² − trace(A)·λ + det(A) = 0. The discriminant (trace² − 4·det) determines whether the eigenvalues are real or complex. When the discriminant is negative, the calculator returns complex conjugate eigenvalues in the form real ± imag·i. For real eigenvalues, it also computes eigenvectors by solving (A − λI)v = 0 using the off-diagonal entries.
For 3×3 matrices, the tool solves the cubic characteristic polynomial λ³ − trace·λ² + cofactorSum·λ − det = 0 using a numerical root-finding method. The algorithm scans the range −100 to +100 in steps of 0.05 and applies bisection with 60 iterations to locate each root to 6-digit precision. Up to three real eigenvalues are reported with approximate eigenvector labels. This approach handles symmetric matrices (which always have real eigenvalues), diagonal matrices, and most common matrices encountered in physics, engineering, and data science applications.
Eigenvalues and eigenvectors appear throughout science and engineering: structural engineers use them to find natural vibration frequencies and mode shapes of structures; data scientists use them in Principal Component Analysis (PCA) to reduce the dimensionality of datasets; control engineers analyze system stability using eigenvalue locations in the complex plane; quantum physicists compute observable quantities as eigenvalues of Hermitian operators. This calculator provides trace, determinant, and step-by-step characteristic equation display for 2×2 matrices to support learning and verification.
Key Features
- Solves 2×2 characteristic equation analytically: λ² − trace·λ + det = 0 with exact discriminant calculation
- Returns real eigenvalues and eigenvectors for positive discriminant, complex conjugates for negative discriminant
- Computes eigenvectors for 2×2 matrices by solving (A − λI)v = 0 using off-diagonal elements
- Numerically finds up to 3 real eigenvalues for 3×3 matrices using bisection root-finding on the cubic characteristic polynomial
- Displays trace (sum of diagonal) and determinant for 2×2 matrices alongside eigenvalue results
- Shows step-by-step characteristic equation, discriminant, and polynomial coefficients for 2×2 case
- Interactive matrix input grid — click any cell to edit, results update instantly
- 100% client-side — no matrix data is uploaded to any server
Frequently Asked Questions
What are eigenvalues and eigenvectors?
For a square matrix A, an eigenvalue λ and its corresponding eigenvector v satisfy the equation Av = λv. This means multiplying the matrix A by the vector v produces a new vector that is simply a scalar multiple (λ) of v — the vector does not change direction, only scale. Eigenvalues reveal fundamental properties of the transformation represented by A: its scaling factors along special directions called eigenvectors.
How is the characteristic equation used to find eigenvalues?
Eigenvalues are found by solving the characteristic equation det(A − λI) = 0, where I is the identity matrix. Subtracting λ from each diagonal entry and setting the determinant to zero produces a polynomial in λ. For a 2×2 matrix this is a quadratic: λ² − trace·λ + det = 0. For a 3×3 matrix it is a cubic polynomial. The roots of this polynomial are the eigenvalues.
What does the discriminant tell us about eigenvalues?
For a 2×2 matrix, the discriminant is trace² − 4·det. If the discriminant is positive, there are two distinct real eigenvalues. If it equals zero, there is one repeated real eigenvalue. If it is negative, the eigenvalues are complex conjugates of the form (trace/2) ± i·√(−discriminant)/2. Complex eigenvalues occur in rotation-like transformations and indicate oscillatory behavior in differential equation systems.
What is the trace of a matrix and how does it relate to eigenvalues?
The trace is the sum of all diagonal elements. It equals the sum of all eigenvalues: trace(A) = λ₁ + λ₂ + ... + λₙ. For a 2×2 matrix: trace = a₁₁ + a₂₂ = λ₁ + λ₂. This relationship is used as a quick check: if you know one eigenvalue, you can find the other by subtracting from the trace.
What is the determinant of a matrix and how does it relate to eigenvalues?
The determinant equals the product of all eigenvalues: det(A) = λ₁ × λ₂ × ... × λₙ. For a 2×2 matrix: det = a₁₁·a₂₂ − a₁₂·a₂₁ = λ₁ × λ₂. A zero determinant means at least one eigenvalue is zero, which means the matrix is singular (non-invertible). The trace-determinant relationship gives both coefficients of the 2×2 characteristic polynomial.
Why does the 3×3 calculator use numerical root-finding instead of an exact formula?
While there is an exact formula for cubic polynomial roots (Cardano's formula), it is numerically unstable and complex to implement correctly for all cases. The numerical bisection method used here reliably finds all real roots in the range −100 to +100 with 6-digit precision, which covers the vast majority of matrices students and engineers encounter. If you need roots outside this range, scale your matrix by a constant factor.
What is Principal Component Analysis (PCA) and how are eigenvalues used?
PCA is a dimensionality reduction technique used in machine learning and statistics. It computes the eigenvalues and eigenvectors of the covariance matrix of a dataset. The eigenvectors (principal components) define the directions of maximum variance, and the eigenvalues represent the amount of variance along each direction. Sorting eigenvalues from largest to smallest lets you select the most informative components and reduce data dimensionality while retaining most of the information.
Can this calculator handle symmetric matrices?
Yes. Symmetric matrices (where A = Aᵀ) always have real eigenvalues and orthogonal eigenvectors, which is the spectral theorem. For a 2×2 symmetric matrix, the discriminant is always non-negative, so the calculator will always return real eigenvalues. For 3×3 symmetric matrices, the numerical root finder reliably finds all three real eigenvalues. Symmetric matrices arise frequently in physics (moment of inertia tensors), statistics (covariance matrices), and structural analysis (stiffness matrices).