Linear Algebra
Study of vectors, matrices, and linear transformations.
Advanced Topics
Eigenvalues and Eigenvectors
What Are Eigenvalues and Eigenvectors?
Imagine you have a transformation that stretches or rotates space. An eigenvector is a special vector that only gets stretched (not rotated), and the eigenvalue tells you how much it's stretched.
Mathematically: \[ A\mathbf{v} = \lambda \mathbf{v} \] where \( \mathbf{v} \) is the eigenvector and \( \lambda \) is the eigenvalue.
Why Are They Useful?
- In physics, they describe vibrations and stability.
- In computer science, they help with search engines and face recognition.
- In economics, they model systems that evolve over time.
How to Find Them
- Solve \( (A - \lambda I)\mathbf{v} = 0 \).
- This usually means solving a polynomial equation for \( \lambda \) (the eigenvalue).
Real-Life Connections
- Google's search algorithm uses eigenvectors!
- Engineers use them to analyze buildings and bridges for safety.
Examples
Finding the directions that stay the same under a transformation.
Analyzing vibrations in bridges using eigenvectors.
In a Nutshell
Eigenvalues and eigenvectors reveal how matrices stretch, squash, or leave vectors unchanged.