eigenfunction

Definition

Let $\mathcal{L}$ be a linear operator acting on a function space. A nonzero function $\psi$ is called an eigenfunction of $\mathcal{L}$ if:

$$ \mathcal{L}\{\psi\} = \lambda \psi $$

where $\lambda$ is a scalar called the eigenvalue corresponding to $\psi$.

Remarks

• Eigenfunctions are the infinite-dimensional analog of eigenvectors.
• The set of eigenfunctions corresponding to a given eigenvalue forms a subspace of the function space.

Example: Linear Shift-Invariant Operators

For linear shift-invariant operators, the complex exponentials $e^{j2\pi \bar{f} \cdot \bar{x}}$ are eigenfunctions:

$$ \mathcal{L}\{e^{j2\pi \bar{f}_0 \cdot \bar{x}}\} = H(\bar{f}_0) e^{j2\pi \bar{f}_0 \cdot \bar{x}} $$

where $H(\bar{f}_0)$ is the eigenvalue (transfer function) associated with frequency $\bar{f}_0$.

This property is why the Fourier_transform is useful for analyzing linear shift-invariant systems: the input can be decomposed into eigenfunctions (complex exponentials), each of which passes through the system multiplied only by a complex constant.

Related Articles

• eigenvalue
• Fourier_transform
• basis