Eigenvalue Calculator
Calculate eigenvalues and eigenvectors of any square matrix. Enter your matrix below and get instant results with detailed step-by-step solutions and characteristic polynomial analysis.
Matrix A
Eigenvalues & Eigenvectors
Characteristic Polynomial
Understanding Eigenvalues & Eigenvectors
Eigenvalue Definition
Scalar λ where Av = λv for non-zero vector v. Represents scaling factor along eigenvector direction.
Eigenvector Direction
Non-zero vector v that only changes in magnitude (not direction) when multiplied by matrix A.
Characteristic Polynomial
det(A - λI) = 0 gives eigenvalues. Polynomial degree equals matrix size.
Diagonalization
If A has n linearly independent eigenvectors, then A = PDP⁻¹ where D is diagonal.
Mathematical Theory & Applications
What are Eigenvalues and Eigenvectors?
For a square matrix A, an eigenvalue λ (lambda) and its corresponding eigenvector v satisfy the equation Av = λv. This means that when matrix A acts on eigenvector v, it only scales the vector by factor λ without changing its direction. Eigenvalues reveal fundamental properties about linear transformations and are central to many areas of mathematics and science.
Eigenvalue equation: Av = λv
Characteristic equation: det(A - λI) = 0
For each λ: (A - λI)v = 0 gives eigenvector v
Historical Development
The concept of eigenvalues emerged from the study of quadratic forms and linear transformations in the 18th and 19th centuries. Leonhard Euler studied rotational motion, while Joseph-Louis Lagrange worked on optimization problems that led to eigenvalue concepts.
The modern theory was formalized by Augustin-Louis Cauchy and later extended by mathematicians like David Hilbert. The term "eigenvalue" comes from German "eigenwert" meaning "characteristic value" or "proper value."
Properties of Eigenvalues
Sum Property
Σλᵢ = tr(A) (trace)
Product Property
Πλᵢ = det(A)
Similarity Invariance
Similar matrices have same eigenvalues
Real Symmetric
Real symmetric matrices have real eigenvalues
Triangular Matrix
Eigenvalues are diagonal elements
Orthogonal Matrix
All eigenvalues have magnitude 1
Real-World Applications
Principal Component Analysis (PCA)
Data dimensionality reduction and feature extraction in machine learning
Quantum Mechanics
Energy levels and quantum states in atomic and molecular systems
Vibration Analysis
Natural frequencies and mode shapes in mechanical and structural engineering
Google PageRank
Web page ranking algorithm based on dominant eigenvector of link matrix
Stability Analysis
System stability in control theory and dynamical systems
Computer Graphics
3D transformations, rotations, and geometric modeling
Frequently Asked Questions
Eigenvalues are scalars (numbers) that represent how much an eigenvector is scaled when multiplied by the matrix. Eigenvectors are the directions that remain unchanged (only scaled) under the matrix transformation. Each eigenvalue has corresponding eigenvector(s).
Yes, eigenvalues can be complex even for real matrices. However, real symmetric matrices always have real eigenvalues. Complex eigenvalues often appear in pairs (complex conjugates) for real matrices and represent rotational components in transformations.
An n×n matrix has exactly n eigenvalues (counting multiplicities). Some eigenvalues may be repeated, and the characteristic polynomial is degree n, so by the fundamental theorem of algebra, it has n roots (eigenvalues).
A zero eigenvalue means the matrix is singular (non-invertible) and has determinant zero. The corresponding eigenvector lies in the null space of the matrix. This indicates the matrix "collapses" some directions to zero.
Principal Component Analysis finds the eigenvectors of the covariance matrix. The eigenvalues represent the variance explained by each principal component. Larger eigenvalues correspond to directions of greater data variation, allowing dimensionality reduction.
In dynamical systems, eigenvalues of the system matrix determine stability. If all eigenvalues have negative real parts, the system is stable. Eigenvalues with positive real parts indicate instability, while those on the imaginary axis represent oscillatory behavior.
For small matrices, we solve the characteristic polynomial. For larger matrices, iterative methods like QR algorithm, power iteration, or Jacobi method are used. These methods are more numerically stable and efficient for high-dimensional problems.