Eigenvalue

9 October 8, 2024

The Eigenvalue is the magnitude of the scaling of the Eigenvector. It is one of the most important mathematical concepts that has been discovered. The eigenvalue is a scalar \(\lambda\) that satisfies the equation \(Mv=\lambda v\). The matrix \(M\) represents the Hamiltonian. The eigenvalues can be used to determine the stability of a system. eigenvalues are the roots of the characteristic equation derived from a set of differential equations.[^5] \(n\) different equations produce \(n\) eigenvalues.[^5] An eigenvalue is a complex number with both real and imaginary parts.[^5] The ratio of the absolute value of the difference between the max and min eigenvalues is the stiffness ratio.[^5]

Rotation Matrix – no real eigenvalues
Fibonacci Sequence – the eigenvalues of the state transition matrix for the Fibonacci sequence is the golden ratio
[[Predator Prey Model]]
Root-Locus Plot of PI Pitch Rate Tracking Controller – uses eigenvalues to determine system stability.
General Eigenvalue Problem
Covariance Matrices – covariance ellipses drawn with eigenvalues.
Boltzmann Transport Equation – uses the eigenvalue of the system.
Piecewise Constant Approximation – uses eigenvalues of the inhour equation
Linear Differential Matrix Equations – defines matrix of eigenvalues
Eigenvalues of State-Space System– The eigenvalues of the state-space system are the eigenvalues of the A matrix.
VPM M16 Tandem – uses eigenvalues plotted agains damping factors
Poles of a System – poles of the system indicate positive or negative eigenvalues
Variational Quantum Eigensolver – quantum algorithm for eigenvalues for large matrices
Hamiltonian
Stability Regions of Runge-Kutta Methods – uses the eigenvalue of the system matrix
Modal Participation Analysis – The square root of the eigenvalues are the modal frequencies
Disadvantages of the Power Method– the power method can only solve a single eigenvalue at a time
Wing Rock – Eigenvalues are used to determine the limit cycles of wing rock phenomenon
Monodromy Matrix – one of the eigenvalues is always 1
Limit Points – occur when a jacobian eigenvalue is 0
Predictor-Corrector Landing Algorithms – solves eigenvalue problems
Metabolic Parameter Estimation – can use eigenvalues for parameter estimation
Stiffness Ratio– ratio of abs of maximum and minimum eigenvalues
Eigenvalues of a Circular Vector Field
BO-105 Dynamic Model – shows eigenvalues of different dynamic modes
Helicopter Rotor Blade Design – uses eigenvalue frequency spectrum
H-alpha decomposition – based on eigenvalue/eigenvector decomposition
PARCS – has a steady-state eigenvalue solution
X-31A Lateral Directional Control Laws – Diagonal values of Q define the lateral/directional eigenvalues of aircraft motion
Convergence Factor – spectral radius is the largest eigenvalue of the iteration matrix $G$
Reactor Kinetics – eigenvalue of the system is the ratio of the time that it takes neutrons to be removed from the system, and the generation time
State-Space Estimators – you can place the eigenvalues anywhere you want if the system is observable
Singular Value Decomposition – eigenvalue-like decomposition for rectangular matrices
Separation Principle – estimator eigenvalues do not change controller eigenvalues
[[Pole Placement for Helicopter Flight Control Systems]]
[[First-Order Systems]] – the absolute value of the minimum real part of the eigenvalues of the system is the approximate rate constant for first-order reactions
[[VAAC FCL0005 Weighting Function]] – explains how eigenvalues of controller is set
[[Sailboat System Identification]] – Eigenvalues of the system matrix are used to evaluate the sailboat model
[[Natural Frequency]] the real and imaginary eigenvalues can give you the natural frequency
[[Damping Ratio]] – the real eigenvalue can give you the damping ratio
[[Half-Life]] – determined from the real-component of the eigenvalue
[[Stability Augmentation System]] – uses the \(B\) matrix to change the system eigenvalues to \(|A|=|A+KB|\)

Sources

Backlinks

[[Complex Numbers]]
[[Eigenvector]]
[[Linear Algebra]]
Poles of a System
[[Uncoupled Dynamic Systems