Does anyone know of an example of a simple physical system where eigenvalues have a physical interpretation? The examples I know of are all in quantum mechanics, which is a bit abstract for me.
However, if you write down the matrix of spring constants for the system and solve for the eigenvalues and eigenvectors of this system you can do something special. If you compress or stretch the molecule along the direction of the one of the eigenvectors then let go, the molecule will continue to vibrate along that same direction. The motion will not spread out to all other degrees of freedom. It will also vibrate with a frequency given by the eigenvalue of that eigenvector.
Additionally, any complex vibration of the system can be broken down into a combination of these independent vibrational modes. This is a simple fact because the eigenvectors form an orthogonal basis for the space.
What lines (through the origin) are mapped back to themselves? Those are the eigenvectors, and the amount by which they're elongated or shortened are the eigenvalues.
So, if we talk about 3d space, and we rotate things - the rotation axis is unchanged. That's an eigenvector (with eigenvalue 1).
If we mirror things - any vector in the mirror plane remains unchanged, that's an eigenvector (with eigenvalue 1), the vector perpendicular to the mirror is unchanged, but flipped, so that's an eigenvector (with eigenvalue -1).
If we dilate everything along the x axis by a factor of 2, say, then the x axis is an eigenvector (with eigenvalue 2), while the y and z axis and any vector in that plane is an eigenvector (with eigenvalue 1). Any other vector is "tilted", so not mapped to itself, so not an eigenvector.
... which means it's probably an imaginary physical system.
Maybe a good physical example is a piece of cloth that warps in 2D, and shrinks, when washed? Eigenvectors would describe the warping (skew, say) and eigenvalues the shrinkage relative to the original warp and weft.
Steve Brunton on YouTube has really good videos on eigenvectors & eigenvalues in context of matrix algebra (and then applied to simultaneous differential equations); https://youtube.com/watch?v=ZSGrJBS_qtc .
Does 'through the origin' imply motion through 'the origin'?
Before describing any system, it's up to you (your "convention") to assert where is the zero-point of your world and in which directions the axes (x,y,z) are pointing.
For instance, in the real world you can choose your 3D coordinate system such that your mirror, as a physical system, keeps the origin untouched (0,0,0) -> (0,0,0). If you decide the origin is a point on the mirror, the equations will be linear: mirror(X) = AX. However if you setup the origin some point far from the mirror, like the center of your eyes, the equations are no longer linear, but affine: mirror(X) = AX+B. Looking at the values of the "AX" part of the system would reveal you the mirroring plane, but now shifted by an offset of "+B" -- the distance between the mirror and your eyes -- because your choice of coordinates was not leaving the origin intact.
Oscillation modes in mass-spring systems. Here is a simple one with 2 masses and 3 springs, so the matrix is only 2-by-2.
https://math24.net/mass-spring-system.html
With more than 2 masses, you don't need to arrange the masses on a line, but you can have a 2d or 3d arrangement, with interconnecting springs. I am sorry I failed to find an example image.
The theory is explained, for example, around page 479 in this Thornton and Marion Classical Dynamics textbook. But you need to read about Lagrangian mechanics (chapter 7) before it makes sense.
https://eacpe.org/app/wp-content/uploads/2016/11/Classical-D...
https://web.archive.org/web/20130728183938/williamcotton.com...
Yep, vibration modes. Vibration frequencies represent their eigenvalues while the shape that the structural system exhibits when subjected to said vibration corresponds to it's eigenvector.
If a structural system is modelled as a linear elastic system it's possible to apply an eigendecomposition of that system and represent it in terms of linear combinations of it's vibration modes/eigenvector, and consequently we can get very accurate representations by using only a hand-full of these eigenvectors.
You know swing sets? We would start to swing back and forth just by moving our legs in a particular frwquencey, and without much effort we could move more and more? It turns out the frequency we moved our legs was the system's vibration frequency/eigenvalue for the vibration modes/eigenvector representing the we swinging back and forth.
Actually, trying to understand how eigenmodes and eigenfrequencies — which I understand well — relate to eigenvalues and eigenvectors.
Yes. The eigenvalues and eigenvectors of an undamped harmonic oscillator are respectively the vibration frequency and vibration mode.
One major class of structural analysis techniques is modal analysis, which determines the vibration modes and corresponding frequencies of specific structural systems subjected to particular boundary conditions.
Principle axes are the axes where a weightless body can rotate around without "wobbling". These axes are orthogonal to each other. If a rigid body has I_1 < I_2 < I_3 moments of inertia, then rotation around the first and third axes is stable and rotation around the second axes is unstable.
[1] https://en.wikipedia.org/wiki/Moment_of_inertia#Inertia_tens...
[2] https://en.wikipedia.org/wiki/Moment_of_inertia#Principal_ax...
You can look up a vibrating drum head (circular membrane) for a simple example.
Say you have two cars linked, with some spring constant;
| --^^-- [c1] --^^-- [c2] --^^--|
where '^^' is a spring and '|' is a wall.
The motion of these cars can be written using the spring forces in the system or, alternately, as the harmonic motion of the undamped system with some natural frequency.
Setting this up as two simultaneous equations (one for each car) and solving for the roots give you the eigenvalues. The natural frequency is the square root of the eigenvalue. In other words, the eigenvalues help you define the natural frequencies which can be used to characterize the motion of the cars in the more complicated spring-mass system.
I like this example because it gives a physical meaning to both eigenvalues and imaginary numbers. It also shows the connection between the sine and cosine and the complex powers of e comes from (since you can show that all three solve the differential equation).
https://drive.google.com/file/d/12SM0SAOvMq166gc8B1b81Y_S7HP...
The third page in particular shows a plot of "amplitude" versus "frequency" to show the "harmonic spectrum of a sawtooth wave". The "frequencies" correspond to the modes of vibration (i.e., sine waves of different frequency), which are the "eigenvectors" in this case. The "amplitudes" are the relative contribution of those vibrations to the overall sound, and these correspond to "eigenvalues".
The article is talking purely about constructing sounds via synthesis, so there's not necessarily a linear system associated with it, but there is a connection. Wave equations represented by linear partial differential equations can often be analyzed as a linear system that has these "modes of vibration" (i.e., series of orthogonal sinusoids at different frequencies). If you were to, for example, model a plucked string (like a guitar), you can model the solution as a weighted sum of eigenvectors (in this case, "modes of vibration" or sinusoids of different frequencies). The "weights" would be the eigenvalues, which determine the spectrum and ultimately the timbre of the sound produced.
That might seem more involved, because it's an infinite-dimensional linear system (i.e., the vectors are functions on a interval, rather than finite lists of numbers). It turns out, though, that the finite-dimensional discretization of an infinite-dimensional linear system (i.e., a partial-differential equation approximated by a finite-dimensional linear system) will sometimes have eigenvectors / eigenvalues that have similar features as the infinite-dimensional case. For example, there are certain finite-difference operators that can be written in matrix form whose eigenvectors will work out to be sampled sinusoids.
I'm not totally sure of the history, but I think a lot of the interest in eigenvectors / eigenvalues as a topic in matrix theory originated from this are (i.e., numerical solutions for partial-differential equations that were used to model physical systems).
The the eigenvectors will be the long term stable state probabilities.
I can't remember the next part exactly, you can look it up in a textbook, but you multiply the matrices KMK, or similar, and the eigenvalues of this are the natural frequencies of the string. The eigenvectors represent the mode shapes, ie the displacement of each mass element.
The same technique is used in Finite Element Analysis to find the modes and modeshapes of complex structures (a car frame, a bridge, etc)
https://en.m.wikipedia.org/wiki/Eigenfunction#Vibrating_stri...
Eigenvectors are the fundamental frequencies of a spring system
source: PW Atkins & RS Friedman, "Molecular Quantum Mechanics 3rd Ed"
> An observable is any dynamical variable that can be measured... in classical mechanics... observables are represented by functions* (such as position as a function of time), in quantum mechanics they are represented by mathematical operators... We shall not in general distinguish between the observable and the operator that represents that observable (such as the position of a particle along the x-axis)"
> "An operator is a symbol for an instruction to carry out some action, an opeeation, on a function...in certain cases, the outcome of an operation is the same function, multiplied by a constant" [that constant being the eigenvalue]
Form: (operator) (eigenfunction) = (eigenvalue) (eigenfunction)
> "An important point is that a general function can be expanded in terms of all the eigenfunctions of an operator, a so-called complete set of functions... then a general function can be expressed as the linear combination [a sum over a complete set of functions, each function having its own coefficient]"
Finally, we get to a practical real-world example (non-QM):
> "...for instance, the straight line g = ax can be recreated over a certain range by superimposing an infinite number of sine functions, each of which is an eigenfunction of the [differentiation] operator, d2/dx2. Alternatively, the same function may be constructed from an infinite number of exponential functions, which are eigenfunctions of d/dx."
Extending this general concept, we can go into the famous Fourier Transform, used widely for all kinds of classical problems in converting waveforms into frequency peaks, such as in musical analysis, electrical engineering, etc. The wiki page on Fourier transforms has a very brief mention of this view, i.e. The Fourier transform decomposes a function into eigenfunctions for the group of translations.
Here's what looks like a deep dive into this approach to the Fourier Transform (2008):
http://www.systems.caltech.edu/dsp/ppv/papers/journal08post/...
The essence of special relativity is that acceleration is a bit weirder than you think. In particular when you accelerate by amount a in some direction x, even after accounting for the usual Doppler shifts you will find that clocks separated from you by that coordinate, appear to tick at the rate 1 + a x/c² seconds per second, where c² is a fundamental constant. Clocks ahead of you tick faster, clocks behind you tick slower (and indeed appear to slow down and approach a ‘wall of death,’ more technically called an ‘event horizon,’ at a distance c²/a. (This effect is called the ‘relativity of simultaneity,’ and it is in some sense the only real prediction of special relativity, as the rest of this comment will show—the other effects of ‘time dilation’ and ‘length contraction’ are second-order and can be derived from this first-order effect.)
This means that the transformation equations for moving into a neighboring reference frame are not the ones that Galileo and Newton proposed,
t' = t
x' = x – v t
t' = t – (v/c²) x
x' = x – v t
p' = (1 + v/c) p
q' = (1 – v/c) q
p' = (1 + φ/N)^N p = e^φ p
q' = (1 – φ/N)^N q = e^{-φ} q
w' = w cosh(φ) – x sinh(φ)
x' = x cosh(φ) – w sinh(φ)
Okay, now for physical interpretation. You asked what physical meaning these eigenvalues and eigenvectors of the Lorentz transformation have, and the answer is this: the eigenvalues (1, 1) and (1, -1) of the Lorentz matrix represent light rays, the p/q description we came up with above was a description of spacetime in terms of light-ray coordinates where we identify an event at a particular place and time with the light rays that it casts, announcing that the event has happened, in the +x and -x directions. On the negative side, these are also the last light rays that were able to touch the event before it happened, so represent "everything it could have possibly known about" -- there is a space between these two "light cones" which is its "relativistic present," the things that anything which was there at the event cannot know about until the future.
The eigenvalues, exp(φ) = sinh(φ) + cosh(φ) = γ + γ β = √[(1 + β)/(1 – β)] and exp(-φ) = √[(1 – β)/(1 + β)], are the Relativistic Doppler shifts of those light rays. Indeed one can read them as e.g. exp(-φ) = 1/γ * 1/(1 + β) , here 1/(1 + β) is the standard Doppler shift formula from nonrelativistic physics and 1/γ is the decrease in frequency due to time dilation.