Eigenvalues and Eigenvectors
Definitions
Eigenvalue Equation
For a matrix , the eigenvalue equation is , which can be rewritten as:
This has a non-trivial solution if and only if .
Characteristic Polynomial
Eigenspace
Algebraic and Geometric Multiplicity
Properties
Examples
Example 1:
Characteristic polynomial:
Eigenvalues:
Eigenvectors:
Example 2: Rotation by in has eigenvalues (complex, no real eigenvectors).
Trace and Determinant
Sum of Eigenspaces
If has distinct eigenvalues, then is the direct sum of the eigenspaces.
Applications
- Stability analysis of dynamical systems
- Quantum mechanics (observable eigenvalues)
- Principal component analysis
- Google's PageRank algorithm
- Vibration analysis in engineering