Eigenvalue

Definition

An eigenvalue is a special scalar associated with a linear transformation or operator, representing the factor by which a corresponding eigenvector is stretched or compressed during that transformation. Eigenvalues play a crucial role in understanding the properties of operators and can be used to analyze stability, dynamics, and even solutions to differential equations.

5 Must Know Facts For Your Next Test

1. Eigenvalues can be found by solving the characteristic polynomial, which arises from the determinant of the operator minus a scalar multiple of the identity.
2. For bounded self-adjoint operators, all eigenvalues are real, and the eigenvectors corresponding to distinct eigenvalues are orthogonal.
3. The spectral theorem states that every bounded self-adjoint operator can be diagonalized using its eigenvalues and eigenvectors.
4. In compact operators, the non-zero eigenvalues form a discrete set that converges to zero, allowing for better control over infinite-dimensional spaces.
5. Eigenvalues have applications in various fields including physics, engineering, and finance, especially in stability analysis and modal analysis.

Review Questions

• How do eigenvalues relate to the stability of dynamical systems and what role do they play in this context?
  • Eigenvalues are critical in assessing the stability of dynamical systems. When examining systems defined by linear differential equations, the sign of the real parts of the eigenvalues indicates whether perturbations will grow or decay over time. If all eigenvalues have negative real parts, the system is stable, as disturbances will diminish. Conversely, if any eigenvalue has a positive real part, the system is unstable since disturbances will grow exponentially.
• Discuss how the spectral theorem applies to bounded self-adjoint operators and what implications this has for their eigenvalues.
  • The spectral theorem for bounded self-adjoint operators states that such an operator can be represented in terms of its eigenvalues and orthogonal eigenvectors. This means that every bounded self-adjoint operator can be diagonalized, which simplifies many calculations and analyses. The implications include that all eigenvalues are real numbers, and eigenvectors corresponding to distinct eigenvalues are orthogonal, leading to better insights into the operator's structure and function.
• Evaluate how compact operators differ in their spectrum compared to general bounded operators and why this distinction is important.
  • Compact operators have unique properties in their spectrum compared to general bounded operators. Specifically, non-zero eigenvalues of compact operators form a discrete set that accumulates only at zero, which provides a clearer understanding of their behavior in infinite-dimensional spaces. This distinction is significant because it allows for more straightforward applications in functional analysis and differential equations, particularly in ensuring convergence properties and understanding stability in physical systems.

Related terms

Eigenvector: An eigenvector is a non-zero vector that only gets scaled (not rotated) when a linear transformation or operator is applied to it.

Spectrum: The spectrum of an operator is the set of all eigenvalues, providing insight into the operator's behavior and characteristics.

Compact Operator: A compact operator is a type of linear operator that maps bounded sets to relatively compact sets, often leading to discrete eigenvalues with specific properties.