Interpretation/significance of the eigenvalues for a system?

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Hi folks! I wasn't sure where to put this... so I put it here! I'm wondering if there is a physical interpretation/significance of the eigenvalues for a system? I've had people tell me things like "they're the basic solutions to the system" but I just don't quite see what they're saying...
 

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it depends what the system describes?
 
nicksauce
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It depends on the system. Example: If you have a matrix describing coupled oscillators, the eigenvalues are the frequencies of the normal modes of the system. If you have a 2D nonlinear system, the eigenvalues of the Jacobian matrix evaluated at the fixed points gives the qualitative nature of the fixed points (attractor, repellor, spiral, etc.).
 
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Hmmm... I'm a first year undergrad, so perhaps the systems in which eigenvalues have significance are yet to be studied? By that I mean that all of the physics and engineering courses I've done so far rely mainly on basic calculus and not differential equations. Does this seem like a possibility as to why the significance of eigenvalues has eluded me so far?
 
nicksauce
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Most likely.
 
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[itex]Ax=\lambda x[/itex] right? Whatever x physically means, for some x the whole matrix acts like a scalar. So, take your favorite system and you can associate a meaning if you think about individual solutions
 
HallsofIvy
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Mathematics is not physics. (Didn't I just say that recently?) Mathematical concepts and equations do not come with an automatic "physical interpretation". What interpretation you give for a particular application depends upon that application
 
Redbelly98
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I'm wondering if there is a physical interpretation/significance of the eigenvalues for a system?
Here's how I've always pictured it ...

A matrix times a vector will in general change both the magnitude and direction of the vector. Except that certain vectors (the eigenvectors) do not change direction; they are simply rescaled. The rescaling factors for these vectors are the eigenvalues.

Equivalently, apply a matrix to the vectors defining the surface of the unit sphere (or unit hypersphere, depending on the dimension of the matrix). The result is an ellipsoid (or hyperellipsoid). The eigenvalues give the lengths of the axes of that ellipsoid.
 
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Here is a brief description of several applications, which can be divided into elementary and advanced:

Geometric: An n x n matrix can be thought of as a geometric transformation in R^n, for example any 2 x 2 matrix is characterized by how it transforms the points on a unit circle. In practice the properties of matrices make are such that the circle will be variously stretched or shrunken (even collapsed) along each of two directions. The directions along which pure stretch or shrink occur are the eigenvectors of the matrix, and the eigenvalues are the scaling factors in those dimensions.

Rotational Dynamics: A general 3 dimensional object without any rotational symmetries will have its moment of inertia be a 3x3 matrix which happens to be symmetric i.e. M = Transpose(M). The eigenvectors of this matrix are called "principle directions" and these are directions passing through the center of mass along which the object can rotate. Each eigenvalues is the moments of inertia for that axis.

Linear Oscillations: Any system of coupled linear oscillators (interconnected springs in n dimensions) can be represented as a matrix for which the components of the eigenvectors are modal amplitudes for each oscillator and the eigenvalues are modal frequencies. This is a corrolary to the next application:

Linear Differential Equations with Constant Coefficients can always be solved exactly with eigenvectors and eigenvalues, and these can be used to represent chemical reactions, oscillations, beam bending, and many more.

Advanced:

The eigenvalue problems of infinite dimensional self-adjoint matrices correspond to 2nd order Linear differential equations with boundary conditions satisfying certain properties. The solutions are functions which correspond to infinite dimensional vectors and most importantly the solution set forms a basis for a function space typically inclusive of the relevant initial conditions. An example is fourier series of sines and cosines which has been applied to wave motion and diffusion and many more.

The standard formulation of quantum mechanics is based on an infinite-dimensional space of vectors and operators. Every physical property corresponds to an operator, and observable values are the eigenvalues and the eigenfunctions are pure states for that property.

The use of eigenvectors really took off with quantum mechanics, but since then there have been enough published applications to fill an entire library shelf section with back volumes of a thick green journal that I have not read all of yet.
 
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