Matrix Eigenvalues Calculator
Calculate eigenvalues and eigenvectors for matrices of various sizes. Includes step-by-step solutions and comprehensive linear algebra explanations.
λWhat are Eigenvalues and Eigenvectors?
Eigenvalues and eigenvectors are fundamental concepts in linear algebra. For a square matrix A, an eigenvector v is a non-zero vector that, when multiplied by A, results in a scalar multiple of itself. The scalar λ (lambda) is called the eigenvalue.
Mathematical Definition
Where A is the matrix, v is the eigenvector, and λ is the eigenvalue
Eigenvalues (λ)
- • Scalar values that represent scaling factors
- • Found by solving det(A - λI) = 0
- • Can be real or complex numbers
- • Determine stability and behavior of systems
Eigenvectors (v)
- • Direction vectors that remain unchanged
- • Found by solving (A - λI)v = 0
- • Form the basis for eigenspaces
- • Used in matrix diagonalization
🔢Calculation Methods
Step-by-Step Process
Characteristic Polynomial
Calculate det(A - λI)
Solve for λ
Find roots of polynomial
Find Eigenvectors
Solve (A - λI)v = 0
Normalize
Scale eigenvectors if needed
2×2 Matrix Example
3×3 Matrix Complexity
- • Cubic characteristic polynomial
- • May require numerical methods
- • Can have complex eigenvalues
- • Multiple eigenvectors possible
🚀Applications of Eigenvalues
🔬 Physics & Engineering
- • Vibration analysis
- • Stability of structures
- • Quantum mechanics
- • Control systems
📊 Data Science
- • Principal Component Analysis
- • Dimensionality reduction
- • Machine learning
- • Image compression
💰 Finance
- • Portfolio optimization
- • Risk analysis
- • Market modeling
- • Option pricing
🌐 Computer Graphics
- • 3D transformations
- • Animation systems
- • Mesh deformation
- • Rendering algorithms
🧬 Biology
- • Population dynamics
- • Gene expression analysis
- • Protein folding
- • Evolutionary models
🔍 Search Algorithms
- • Google PageRank
- • Recommendation systems
- • Network analysis
- • Graph algorithms
📐Important Properties and Theorems
🔹 Trace Property
Sum of eigenvalues equals the trace of the matrix
🔹 Determinant Property
Product of eigenvalues equals the determinant
🔹 Similarity Invariance
Similar matrices have the same eigenvalues
🔸 Symmetric Matrices
Always have real eigenvalues and orthogonal eigenvectors
🔸 Positive Definite
All eigenvalues are positive
🔸 Diagonalization
Matrix is diagonalizable if it has n linearly independent eigenvectors
⚡Computational Methods
Direct Methods
- • Characteristic Polynomial: Exact for small matrices
- • Analytical Solutions: 2×2 and some 3×3 matrices
- • Symbolic Computation: Computer algebra systems
Challenges
- • Numerical Stability: Sensitive to rounding errors
- • Complex Eigenvalues: Require complex arithmetic
- • Large Matrices: Computational complexity O(n³)
Iterative Methods
- • Power Method: Dominant eigenvalue
- • QR Algorithm: All eigenvalues
- • Jacobi Method: Symmetric matrices
Software Libraries
- • LAPACK: Linear algebra routines
- • NumPy/SciPy: Python scientific computing
- • MATLAB: Built-in eig() function