Eigenvalue Calculator 2026
Instantly compute eigenvalues, trace, determinant, and characteristic polynomials for 2×2 and 3×3 matrices. Perfect for linear algebra, physics, and engineering students.
Matrix Input
Define your square matrix to solve for eigenvalues
Select the size of the square matrix to analyze
Enter the numerical values for each cell of the matrix
Linear Algebra Solution
Matrix properties & eigenvalues
Enter your matrix elements above and click Solve for Eigenvalues to reveal the characteristic polynomial, trace, determinant, and eigenvalues.
Matrix Properties & Definitions
Core concepts and mathematical properties related to eigenvalues and eigenvectors in linear algebra.
| Property | Symbol | Definition / Formula | Significance |
|---|---|---|---|
| Trace | tr(A) | Sum of main diagonal elements | Equals the sum of all eigenvalues |
| Determinant | det(A) | Scaling factor of the linear transformation | Equals the product of all eigenvalues |
| Eigenvalue | λ | Scalar satisfying Av = λv | Represents the factor by which the eigenvector is scaled |
| Eigenvector | v | Non-zero vector satisfying Av = λv | Direction that remains unchanged by the transformation |
| Characteristic Polynomial | p(λ) | det(A – λI) = 0 | The roots of this polynomial are the eigenvalues |
Eigenvalue & Matrix FAQ
Learn more about linear algebra concepts, matrix transformations, and eigenvalue properties.
An eigenvalue (λ) and its corresponding eigenvector (v) are special scalars and non-zero vectors that satisfy the equation Av = λv, where A is a square matrix. Geometrically, an eigenvector points in a direction that is stretched or compressed by the linear transformation represented by the matrix, and the eigenvalue is the factor by which it is scaled.
To find the eigenvalues of a matrix, you must solve the characteristic equation det(A – λI) = 0, where I is the identity matrix of the same size. This results in a polynomial equation in terms of λ. The roots of this polynomial are the eigenvalues. For a 2×2 matrix, this is a quadratic equation; for a 3×3, it is a cubic equation.
The characteristic polynomial of an n×n matrix A is defined as p(λ) = det(A – λI). It is a polynomial of degree n, and its roots are exactly the eigenvalues of the matrix. The coefficients of this polynomial are deeply connected to matrix invariants like the trace and determinant.
Yes, a real matrix can absolutely have complex eigenvalues. This typically occurs when the matrix represents a rotation or an oscillatory system. Complex eigenvalues always appear in conjugate pairs (e.g., a + bi and a – bi) for real matrices, reflecting the rotational nature of the transformation in a 2D subspace.
If a matrix has an eigenvalue of zero, it means that the determinant of the matrix is also zero (since the determinant is the product of all eigenvalues). Such a matrix is called ‘singular’ or ‘non-invertible’, meaning it squashes at least one dimension of the vector space into a lower dimension, and its columns are linearly dependent.