Difference Between Eigenvalue, Eigenvector and Eigenfunction
1. Eigenvalue and Eigenvector (Linear Algebra)
Definition
For a square matrix \( A \):
\[ A \mathbf{v} = \lambda \mathbf{v} \]
- \( \mathbf{v} \neq 0 \) → Eigenvector
- \( \lambda \) → Eigenvalue
- \( A \) → Linear transformation
The matrix scales the vector without changing its direction.
Finding Eigenvalues
\[ (A - \lambda I)\mathbf{v} = 0 \]
For non-trivial solution:
\[ \det(A - \lambda I) = 0 \]
This is the characteristic equation.
Numerical Example
\[ A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} \]
Step 1: Eigenvalues
\[ \det(A - \lambda I) = (2-\lambda)(3-\lambda) \]
\[ \lambda_1 = 2, \quad \lambda_2 = 3 \]
Step 2: Eigenvectors
For \( \lambda = 2 \):
\[ \mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \]
For \( \lambda = 3 \):
\[ \mathbf{v}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]
The matrix scales these vectors without changing direction.
2. Eigenfunction (Signals & Systems)
Definition
\[ T\{x(t)\} = \lambda x(t) \]
- \( x(t) \) → Eigenfunction
- \( \lambda \) → Eigenvalue
- Function shape remains unchanged
Example: LTI System
\[ y(t) = x(t) * h(t) \]
If input:
\[ x(t) = e^{j\omega t} \]
Output:
\[ y(t) = H(j\omega) e^{j\omega t} \]
- Eigenfunction = \( e^{j\omega t} \)
- Eigenvalue = \( H(j\omega) \)
Numerical Example
System:
\[ y(t) = 3x(t) \]
If input:
\[ x(t) = \sin(5t) \]
Output:
\[ y(t) = 3\sin(5t) \]
- Eigenfunction = \( \sin(5t) \)
- Eigenvalue = 3
3. Key Differences
| Concept | Used In | Mathematical Form | Object Type | Meaning |
|---|---|---|---|---|
| Eigenvector | Linear Algebra | \( A\mathbf{v}=\lambda\mathbf{v} \) | Vector | Direction unchanged |
| Eigenvalue | Linear Algebra & Systems | Scaling factor | Scalar | Amount of scaling |
| Eigenfunction | Systems | \( T\{x\}=\lambda x \) | Function | Shape unchanged |
4. Summary
\[ \boxed{\text{Eigenvector → vector unchanged in direction}} \]
\[ \boxed{\text{Eigenfunction → function unchanged in shape}} \]
\[ \boxed{\text{Eigenvalue → scaling factor in both cases}} \]