🔢 Eigenvalue and Eigenvector Calculator
Find eigenvalues λ and eigenvectors v such that Av = λv
How to Use This Calculator
Enter Matrix
Input all elements of your 2×2 square matrix.
Calculate
Click to find eigenvalues and eigenvectors by solving Av = λv.
Review Results
See eigenvalues, corresponding eigenvectors, trace, and determinant.
Formula
A v = λ v
Where λ is eigenvalue and v is eigenvector
Finding Eigenvalues:
Solve det(A - λI) = 0
For 2×2: det(A - λI) = λ² - trace(A)λ + det(A) = 0
λ = (trace ± √(trace² - 4·det)) / 2
Finding Eigenvectors:
For each eigenvalue λ, solve (A - λI)v = 0
Find non-zero solution v
Properties:
- Trace(A) = λ₁ + λ₂ + ... + λₙ
- det(A) = λ₁ × λ₂ × ... × λₙ
- Eigenvectors are not unique (any scalar multiple is also eigenvector)
About Eigenvalue and Eigenvector Calculator
The Eigenvalue and Eigenvector Calculator finds eigenvalues λ and eigenvectors v of a square matrix A such that Av = λv. Eigenvalues are fundamental scalars that characterize the matrix, and eigenvectors are directions that are preserved under multiplication by A.
When to Use This Calculator
- Linear Algebra: Understand matrix structure and properties
- Differential Equations: Solve systems of linear ODEs
- Dynamical Systems: Analyze stability and behavior
- Principal Component Analysis: Find principal directions in data
- Quantum Mechanics: Energy levels and eigenstates
- Vibration Analysis: Natural frequencies and modes
Why Use Our Calculator?
- ✅ Complete Solution: Shows eigenvalues and eigenvectors
- ✅ Normalized Vectors: Eigenvectors are normalized
- ✅ Properties: Displays trace and determinant
- ✅ Handles Complex: Shows complex eigenvalues when they occur
- ✅ Educational: Helps understand eigenstructure
- ✅ Free: No registration required
Key Concepts
- Eigenvalue: Scalar λ such that Av = λv for some non-zero v
- Eigenvector: Non-zero vector v such that Av = λv
- Characteristic Polynomial: p(λ) = det(A - λI)
- Trace: Sum of eigenvalues = sum of diagonal elements
- Determinant: Product of eigenvalues
- Geometric Multiplicity: Number of linearly independent eigenvectors for an eigenvalue
Applications
Diagonalization: If A has n independent eigenvectors, A = PDP⁻¹ where D contains eigenvalues and P contains eigenvectors.
Matrix Powers: Aⁿ = P Dⁿ P⁻¹, making it easy to compute high powers of A.
Stability: In dynamical systems, eigenvalues determine stability: negative real parts mean stable.
Frequently Asked Questions
What are eigenvalues and eigenvectors?
Eigenvalues λ are scalars and eigenvectors v are non-zero vectors such that Av = λv. Eigenvectors point in directions that are preserved under multiplication by A.
How do I find eigenvalues?
Solve det(A - λI) = 0. This gives the characteristic polynomial. For 2×2: λ² - trace(A)λ + det(A) = 0.
Can eigenvalues be complex?
Yes, for real matrices, complex eigenvalues come in conjugate pairs. For example, if a+bi is an eigenvalue, then a-bi is also an eigenvalue.
Are eigenvectors unique?
No, eigenvectors are not unique. If v is an eigenvector, then any scalar multiple cv (c ≠ 0) is also an eigenvector for the same eigenvalue.
What's the relationship between trace/determinant and eigenvalues?
Trace(A) = sum of eigenvalues, det(A) = product of eigenvalues. These relationships hold for any square matrix.
Can a matrix have zero eigenvalues?
Yes! If det(A) = 0, then at least one eigenvalue is zero. This means the matrix is singular (not invertible).
What if eigenvalues are repeated?
Repeated eigenvalues can have one or more eigenvectors. If there are fewer eigenvectors than the multiplicity, the matrix is "defective" and cannot be diagonalized.