Math
Explainer
Eigenvectors: The Directions a Matrix Keeps
Some vectors survive a matrix transformation without turning at all.
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Lesson notes
Key points
- A square matrix A transforms vectors, usually rotating and stretching them into new directions.
- An eigenvector is a nonzero vector whose direction is unchanged, only scaled by a number called its eigenvalue.
- Picture it: the matrix stretches v along its own line, landing on λ times v.
- Eigenvectors reveal natural axes, used in vibration analysis, stability of systems and principal component analysis.
Definitions
- Linear transformation
- A rule x → Ax that preserves addition and scalar multiplication
Takeaway: Eigenvectors keep their direction; eigenvalues tell how much they stretch.
Quick check: If Av = 3v with v ≠ 0, what is 3?
- An eigenvalue of A
- The determinant of A
- An eigenvector of A
Linear mapEigenvectorSame lineApplications
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