Eigenvector Calculator
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What are Eigenvectors and Eigenvalues?
Eigenvectors are special vectors that maintain their direction after a linear transformation is applied. When a matrix multiplies an eigenvector, the result is a scaled version of the same vector.
Eigenvalues are the scaling factors by which eigenvectors are stretched or compressed during the transformation.
Mathematical Definition
For a square matrix A, a non-zero vector v is an eigenvector if:
A·v = λ·v
Where λ (lambda) is the eigenvalue corresponding to the eigenvector v.
How to Find Eigenvectors and Eigenvalues
The process involves:
- Solving the characteristic equation: det(A - λI) = 0
- Finding the eigenvalues (λ) that satisfy this equation
- For each eigenvalue, solving (A - λI)v = 0 to find the corresponding eigenvectors
Applications of Eigenvectors
Eigenvectors have numerous applications in various fields:
- Physics: Vibration analysis, quantum mechanics
- Computer Science: Principal Component Analysis (PCA), Google's PageRank algorithm
- Engineering: Stability analysis, control systems
- Economics: Input-output models, portfolio optimization
Frequently Asked Questions
What is the difference between eigenvectors and eigenvalues?
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Eigenvectors are the vectors that maintain their direction after a transformation, while eigenvalues are the scaling factors that indicate how much the eigenvectors are stretched or compressed.
Can a matrix have multiple eigenvectors?
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Yes, a matrix can have multiple eigenvectors. In fact, an n×n matrix can have up to n linearly independent eigenvectors, though this is not always the case.
What does it mean if an eigenvalue is zero?
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If an eigenvalue is zero, it means the matrix is singular (non-invertible). The corresponding eigenvector lies in the null space of the matrix.
Can eigenvectors be complex numbers?
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Yes, for matrices with real entries, complex eigenvalues and eigenvectors can occur in conjugate pairs. This happens when the characteristic equation has complex roots.