[HN Gopher] What is an eigenvalue?
___________________________________________________________________
What is an eigenvalue?
Author : RafelMri
Score : 164 points
Date : 2022-11-08 10:29 UTC (12 hours ago)
(HTM) web link (nhigham.com)
(TXT) w3m dump (nhigham.com)
| raydiatian wrote:
| A far easier to digest primer on eigenvalues is available from
| 3Blue1Brown [+]. His presentation format is undeniably
| approachable, so much so that I think you could probably use it
| to teach linear algebra and eigenvectors to 9-year olds.
|
| [+] https://youtu.be/PFDu9oVAE-g
| mpaepper wrote:
| I explained it in a more coding oriented style here:
| https://www.paepper.com/blog/posts/eigenvectors_eigenvalues_...
| [deleted]
| hdjjhhvvhga wrote:
| For a completely different approach, see this answer:
|
| https://www.reddit.com/r/explainlikeimfive/comments/1avwm7/c...
| oifjsidjf wrote:
| I was blown away in my Digital Signal Processing (DSP) class that
| eigen "values" exist for certain systems in the form of "waves".
|
| Basicaly you put in a wave made from multiple sine and/or cosine
| waves through some function f(x) and the output is STILL a wave,
| though its frequency, amplitude and phase might change.
|
| Technicaly if I remember correctly this applies to all complex
| exponentials, since those can be rewritten in the form of e^(ix)
| = cosx + i*sinx.
|
| This formula also beatifuly shows how rotations and the complex
| exponentials are connected.
|
| So basicaly you don't just have eigen values, eigen vectors: you
| also have eigen FUNCTIONS (sine and cosine above are the eigen
| functions of f(x)).
|
| DSP basicaly revolves arounds functions that don't "corrupt"
| wave-like inputs (wave in -> wave out).
| WastingMyTime89 wrote:
| I'm not sure I understand but it seems to me you are just
| talking about eigenvalues in C.
|
| That's interesting but not particularly remarkable because
| eigenvalues are defined for linear transformations of any
| vector space over a field.
| mrfox321 wrote:
| https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
| bobbylarrybobby wrote:
| If you're into eigenfunctions, pick up any textbook on quantum
| mechanics. The hamiltonian is a linear operator whose
| eigenfunctions are the stationary states of the system
| ("stationary" because an isolated system in a stationary state
| will never leave that state) and whose eigenvalues are the
| observable values of the energy of the system. In general,
| there is a correspondence between observable quantities and
| Hermitian operators on wavefunctions: "measurement" is the
| application of a Hermitian operator to the wavefunction, and
| the values you may observe are the eigenvalues of the operator.
| So, for instance, energy is quantized in some systems because
| their hamiltonian has discrete eigenvalues.
| sverona wrote:
| In fact functions are just infinite-dimensional vectors. Almost
| all of the theory goes through unchanged. This is the basic
| idea of functional analysis.
| constantcrying wrote:
| Notably, one _very_ important part which does not go through
| is that for mappings between infinite dimensional spaces
| linearity does not imply continuity. (E.g. a series of
| functions bounded by a constant can have arbitrarily large
| derivatives)
|
| A large part of functional analysis is dealing with that fact
| and its implication for PDEs.
| OJFord wrote:
| Or rather size-of-their-domain-dimensional?
| ravi-delia wrote:
| If you want, but you can do better. I believe, for
| instance, that at least continuous functions on the reals
| have a countable basis. Might even be as strong as
| measurable, not sure about that. That's how, for instance,
| fourier transforms work.
| constantcrying wrote:
| The fourier Transform essentially means that every L^2
| (the square of the function has a finite integral) is
| completely "described" by an l^2 series (a series of
| numbers whose sum of squares converges), which is about
| the greatest piece of magic in mathematics. One _very_
| important piece here is that the term "function" is
| somewhat of a lie (since the result couldn't be true if
| it weren't).
|
| The result for measurable _functions_ (not almost
| functions) shouldn 't be true (I think). I am not even
| sure it is true for L^1 almost functions.
| [deleted]
| c-baby wrote:
| Maybe I'm missing what's interesting about this, but a function
| like f(z) = 5z + 2 would output a wave with changed amplitude
| and phase when z = sin(x). That doesn't seem that interesting
| to me, so f(z) must have some other interesting properties?
| oifjsidjf wrote:
| Honestly I forgot the details, but basical the ENTIRE field
| of DSP stands on this fact.
|
| Basicaly there exist some functions into which you can feed
| in sound waves and the output is guaranteed to still be a
| sound wave.
|
| If you'd feed in a sound wave and if the function would
| corrupt it you would not be able to do any digital signal
| processing, since the output must be a wave.
|
| Sound(wave) in -> Sound(wave) out, guaranteed to always be
| true.
| lisper wrote:
| > there exist some functions into which you can feed in
| sound waves and the output is guaranteed to still be a
| sound wave.
|
| That in and of itself does not seem like a particularly
| insightful observation. It's just _obvious_ that such
| functions exist. I can think of three of them off the top
| of my head: time delay, wave addition, and multiplication
| by a scalar. There must be something more to it than that.
| brendanclune wrote:
| In math, the obvious things aren't always true and the
| true things are often not obvious.
|
| Trivially, the identity f(x) = x satisfies the guarantee
| as well. What amounts to insightful observation is the
| definition and classification of these functions. In
| exploring their existence in various forms, we can begin
| to understand what properties these functions share.
|
| So the interesting part is not that this class of
| function _exists_, because of course it does! Your
| intuition has led you to three possible candidates. But
| if we limit ourselves to only the functions that satisfy
| the condition _wave-in implies wave-out_ what do they
| look like as a whole? What do these guarantees buy us if
| we _know_ the result will be a wave? For example, f(g(x))
| is also guaranteed to be _wave-in-wave-out_. Again, maybe
| obvious, but it's a building block we can use once we've
| proved it true.
| c-baby wrote:
| canadianfella wrote:
| geysersam wrote:
| It's even worse than you describe it!
|
| f needs to be linear, but the function in your example is not
| linear.
|
| However, there are quite interesting linear functions.
| Example: f(x(t)) = x(t-2) + 4dx/dt - \int_0^t 2x(s) ds
| c-baby wrote:
| 5z + 2 is linear?
| lp251 wrote:
| affine, not linear. describes a line that doesn't go
| through the origin. that pesky shift breaks linearity
|
| 5(2z) + 2 != 2(5z + 2)
| topaz0 wrote:
| Small correction: eigenfunctions are analogous to eigenvectors,
| not eigenvalues. In fact they _are_ eigenvectors, in the sense
| that they are vectors in a vector space of functions (or some
| restricted set of functions, e.g. continuous functions or
| periodic functions).
| [deleted]
| [deleted]
| adamnemecek wrote:
| All these points fail to mention that they are fundamentally
| self-relationship
|
| Lawvere's fixed point theorem is I think the best formulation of
| the idea
| https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theore...
|
| I've been putting together a brain dump on the topic
|
| https://github.com/adamnemecek/adjoint/
|
| Join the discord
|
| https://discord.gg/mr9TAhpyBW
| kiernanmcgowan wrote:
| Eigen-things can also be thought of "fixed values" of a "thing"
| transformation.
|
| For example - the eigenfunction of a derivative is e^x since when
| you run the derivative function on e^x you get.... e^x
| Tainnor wrote:
| They don't have to be fixed, a scalar multiple is allowed too.
| e^ax is an eigenfunction of the derivative too.
| marviel wrote:
| After watching the 3blueonebrown video linked below in the
| comments, I'm inclined to agree with you -- nice way of putting
| it.
| [deleted]
| AdamH12113 wrote:
| 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.
| 6gvONxR4sf7o wrote:
| Eigenvalues of covariance matrices are a famous example. You
| can get PCA from it.
| cat_man wrote:
| There are a couple of other comments that have mentioned
| oscillation modes, vibrations, etc. The first 7 pages of this
| series on sound synthesis might help give an idea of where
| these might come from:
|
| 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).
| CamperBob2 wrote:
| Wow, that's an awesome introduction to music synthesis.
| Bookmarking for future referral to others.
| [deleted]
| idiotsecant wrote:
| Get a sheet of rubber. Grab it in both hands and stretch it.
| Inspect your sheet and find a line on the sheet that you could
| draw in with a marker and when you stretched the sheet the line
| would grow and shrink, but would not change what it was
| pointing at (probably a line from one of your hands to the
| other, in this simple example) That is an eigenvector of your
| sheet stretching transformation. The eigenvalue is how hard
| you're stretching the sheet.
| einpoklum wrote:
| I'll upvote any post beginning with "get a sheet of rubber"
| :-)
| olddustytrail wrote:
| Except the "how to have sex with a leopard" post, because
| describing rubber as "protection" in such circumstances is
| really stretching.
| antegamisou wrote:
| Most are introduced to the interplay between physics and linear
| algebra through the study of the mass - spring system where the
| type (real or complex), sign and amount of the eigenvalues
| determine its behavior and stability. For example, complex
| eigenvalues with positive real part indicate an unstable, or
| chaotic, in terms of amplitude convergence oscillation.
| johnbcoughlin wrote:
| The speed of sound is the eigenvalue of a particular matrix
| (the "flux Jacobian") in the Euler equations, the 5-component
| system of partial differential equations that describe gas
| dynamics.
| bodhiandphysics wrote:
| Take an airplane... it's dynamics are described by a series of
| differential equations. We want to know if it's stable! If the
| wife values of the dynamics are real and greater than 1 it's
| unstable. If the eigenvalues are complex and have a modulus
| greater than 1 it will oscillate instability. If one is equal
| to one, it will cause everyone to vomit.
| _spduchamp wrote:
| I read an interview with Australian wire music composer Alan
| Lamb that a stringed instrument with multiple overtones
| vibrating on the string can be analyzed by breaking down the
| vibration into eigenvalues, but I've never found any reference
| material that explain that. I'm wondering if he was referring
| to FFT.
| alanbernstein wrote:
| Complex exponentials are the eigenfunctions of the Fourier
| transform. In other words, frequency component values are the
| eigenvalues.
|
| https://en.m.wikipedia.org/wiki/Eigenfunction#Vibrating_stri.
| ..
| contravariant wrote:
| That makes no sense, the Fourier transform of a complex
| exponential is a delta function.
| alanbernstein wrote:
| Hmm, you're right, that should have been obvious. Thanks
| for the correction.
| sfpotter wrote:
| See my other reply.
| auxym wrote:
| If you discretize the string into a bunch of tiny masses,
| linked together by a bunch of tiny springs, you can build a
| mass Matrix (diagonal) M and a Stiffness matrix K (element ij
| = stiffness of spring that links mass I and mass j).
|
| 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)
| bodhiandphysics wrote:
| Also in computer science.. a web sites page rank is the
| eigenvalue of the connectivity matrix.
| cscheid wrote:
| That's not true. The page rank is read from the eigenvector,
| and is the value associated with the given vertex (ie web
| page). There are as many page rank values as there are web
| pages, but only one eigenvector from which to read: the
| dominant eigenvector of the transition matrix, which is the
| one with the largest eigenvalue. So, only a single eigenvalue
| for the entire pagerank computation.
| bodhiandphysics wrote:
| You're right!!! Acch...
| contravariant wrote:
| If i recall correctly you can represent a harmonic oscillator
| as a linear differential equation with a 2x2 matrix. The
| imaginary part of the eigenvalues of this matrix correspond to
| the angular frequency of the oscillator.
|
| 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).
| hn_throwaway_99 wrote:
| Don't know if this counts as a "physical" system, but Google's
| original PageRank algorithm famously uses eigenvectors and
| eigenvalues:
| https://math.stackexchange.com/questions/936757/why-is-pager...
| williamcotton wrote:
| I have a brief overview of eigenvectors as a 2D shear
| transformation in this overview of PageRank:
|
| https://web.archive.org/web/20130728183938/williamcotton.com.
| ..
| hn_throwaway_99 wrote:
| Oooh, this is great! Thanks very much.
| FabHK wrote:
| Take a linear map from some space to itself, and ask:
|
| 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.
| MichaelZuo wrote:
| What does 'through the origin' mean in a physical system?
| pbhjpbhj wrote:
| It means it doesn't matter where it is: you can choose the
| origin, ie the point you measure from, it is arbitrary. Or
| another way of saying that is you can move the system to a
| different set of coordinates and it works in the same way.
|
| ... 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 .
| MichaelZuo wrote:
| Okay, so that explains 'the origin'.
|
| Does 'through the origin' imply motion through 'the
| origin'?
| mdup wrote:
| It means the eigenvalues will only give you information
| about the system relatively to the center of that system.
|
| 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.
| DennisP wrote:
| When you're rotating something, the axis of rotation.
| That's the point that doesn't change in rotation ("maps to
| itself").
| theGnuMe wrote:
| Center of mass; the object itself.
| montecarl wrote:
| I think a system of springs is a good example. I think having a
| bunch of springs hooked together is a bit abstract so let's
| instead think of a molecule and model the bonds between the
| atoms as springs. If you were to squeeze this molecule together
| or try to pull it apart and then let go, it would vibrate in
| some complex way. By complex I mean that it wouldn't just
| bounce back along the direction that you compressed or
| stretched it.
|
| 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.
| sampo wrote:
| > Does anyone know of an example of a simple physical system
| where eigenvalues have a physical interpretation?
|
| 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...
| xhkkffbf wrote:
| Think of a fun house mirror that for the sake of this example
| make you look twice as tall but 20% skinnier. This can be
| modeled by a two-by-two matrix with eigenvalues of 2 and 0.8.
| (Indeed, it will have them on the diagonals which makes it
| easier to study.)
| syrrim wrote:
| The vibration of a bell (say) could be modelled by a matrix,
| with a state vector to represent position, velocity, and
| acceleration, and the matrix modelling the differential
| equations describing their evolution over time. The
| eigenvectors represent a basis of the system, so that we can
| describe any potential state vector as a sum of eigenvectors.
| If we do so, then each step of the system can be modelled by
| multiplying each of these eigenvectors by its corresponding
| eigenvalue. If an eigenvalue happens to be complex, then we can
| describe it in phasor form as the product of an amplitude and a
| angle. The amplitude tells us how it will decay (or amplify)
| over time. The angle tells us the frequency of oscillation, and
| thus the note that the bell will typically sound.
| qntty wrote:
| Natural frequencies of mechanical systems are eigenvalues of
| it's equation of motion.
| _spduchamp wrote:
| Any references to help me unpack what you just said there?
| sfpotter wrote:
| Things that vibrate have natural modes of vibration. A
| particular vibrational pattern can be decomposed into a
| time-varying linear combination of these modes. The modes
| of vibration are eigenfunctions and the frequencies at
| which they vibrate are the square root of the corresponding
| eigenvalues.
|
| You can look up a vibrating drum head (circular membrane)
| for a simple example.
| bumby wrote:
| _Theory of Vibration with Applications_ by William Thompson
| and Marie Dillon Dahleh.
|
| 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.
| wolfi1 wrote:
| Landau/Lifshitz: "Mechanics" has a chapter on small
| oscillations
| pvg wrote:
| That's more of a further packing than an unpacking.
| Although that totally should be an expression for things
| that go on for too long: "Can you pack this for me
| please"
| planede wrote:
| Moment of inertia tensor [1], and principal axes of rotation
| [2].
|
| 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...
| atty wrote:
| When dealing with the Schrodinger equation, the eigenvalues are
| the energy levels of the quantum system.
| ok123456 wrote:
| Markov probability matrix where the entries are probabilities
| of some physical event happening.
|
| The the eigenvectors will be the long term stable state
| probabilities.
| litoE wrote:
| Not quite. For a Markov probability matrix, 1 is always an
| eigenvalue, and all other eigenvalues are less than or equal
| to 1. For each eigenvalue that is equal to 1 you get a long
| term stable state probability. These distributions contain
| disjoint subsets of the states, and the system will converge
| to one of those subsets, depending on the initial state. The
| eigenvalues that are strictly less than 1 do not add any
| information to the long term state of the system. See
| Stochastic Processes and Their Applications, V4 (1976) pages
| 253-259. I wrote it while still in grad school.
| cscheid wrote:
| The values associated with each vertex on the _dominant
| eigenvector_ (the eigenvector associated with the dominant
| eigenvalue) are the long-term stable state probabilities.
| That's from a single eigenvector, not "the eigenvectors".
| ok123456 wrote:
| Yeah each one is an eigenmode of the system. That's what I
| meant.
| crdrost wrote:
| So my actual favorite first example is to do this with
| Fibonacci numbers as a linear recurrence relation, but that's
| not really a "physical" interpretation. Let me give you my
| favorite physical one:
|
| 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
|
| but slightly modified to (to first order in v, so only
| considering small velocity changes) t' = t -
| (v/c2) x x' = x - v t
|
| where _w_ = c _t_ is a measure of time in units of distance
| using this fundamental constant. How do we generalize and get
| the full solution? We can do it by looking in the eigenvector
| basis. Consider new coordinates _p_ = _x_ - c _t_ and _q_ = _x_
| + c _t_ , given any ( _x_ , _t_ ) you can find a unique ( _p_ ,
| _q_ ) which describes it and if you want to get back those
| values you would say _x_ = ( _p_ + _q_ )/2, _t_ = ( _q_ - _p_
| )/(2 c). But feed these magical coordinates that come from
| eigenvectors into the above transform and it "diagonalizes",
| p' = (1 + v/c) p q' = (1 - v/c) q
|
| and therefore if you want to make a big change in "velocity" c
| ph (here instead ph turns out to be "rapidity") out of N
| smaller changes, you can repeat this transform N times with
| little boosts by v/c = ph/N, and you will stitch together the
| full Lorentz transform out of little first-order Lorentz
| transforms: p' = (1 + ph/N)^N p = e^ph p
| q' = (1 - ph/N)^N q = e^{-ph} q
|
| Transforming back and using the hyperbolic sine and cosine,
| sinh(x) = (e^x - e^{-x})/2, cosh(x) = (e^x + e^{-x})/2, the
| full formula is w' = w cosh(ph) - x sinh(ph)
| x' = x cosh(ph) - w sinh(ph)
|
| where _w_ = c _t_ is a simple time-in-units-of-meters
| coordinate. Usually we denote cosh(ph) = g, sinh(ph) = g b,
| which gives this the more familiar form you 'll find in
| textbooks, and the identity cosh2x = 1 + sin2x gives a formula
| g = 1/[?](1 - b2) for the latter... but this 'rapidity form' is
| in some ways more elegant. Anyway, point stands, from the
| "first-order" transform you can derive the "full" transform
| just by building any large velocity change out of an infinite
| number of infinitesimal velocity changes, and this is the
| source of the factor g which describes time dilation and length
| contraction.
|
| 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(ph) = sinh(ph) + cosh(ph) = g + g b =
| [?][(1 + b)/(1 - b)] and exp(-ph) = [?][(1 - b)/(1 + b)], are
| the Relativistic Doppler shifts of those light rays. Indeed one
| can read them as e.g. exp(-ph) = 1/g * 1/(1 + b) , here 1/(1 +
| b) is the standard Doppler shift formula from nonrelativistic
| physics and 1/g is the decrease in frequency due to time
| dilation.
| simplotek wrote:
| > Does anyone know of an example of a simple physical system
| where eigenvalues have a physical interpretation?
|
| 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.
| dr_dshiv wrote:
| Does this relate to the normal modes or eigenmodes of a
| system?
|
| Actually, trying to understand how eigenmodes and
| eigenfrequencies -- which I understand well -- relate to
| eigenvalues and eigenvectors.
| simplotek wrote:
| > Does this relate to the normal modes or eigenmodes of a
| system?
|
| 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.
| [deleted]
| billfruit wrote:
| I do think, the term eigenvalue is rather opaque, and should be
| replaced by a more plain-english terminology that readily conveys
| its meaning.
| Tainnor wrote:
| It used to be called "proper value" in English (you can still
| find that in old textbooks), but the (semi-)German word has
| basically entirely replaced it.
| st_goliath wrote:
| As a native German speaker, I don't understand your problem.
| It's very much not opaque, plain terminology that easily
| conveys meaning. ;-)
|
| Perhaps, we should compromise and name it after Leonhard Euler?
| That should clear up the confusion.
| NotYourLawyer wrote:
| "Characteristic value." I guess that's a little better
| actually.
| frozenlettuce wrote:
| In romance languages you have
| autovettore/autovetor/autovectore, as in "self-vector"
| hdjjhhvvhga wrote:
| Can you propose one?
| billfruit wrote:
| Linear scaling factor?
|
| Scale of aspect?
|
| Aspect factor?
|
| Scale along Axis?
|
| Axial scaling factor?
|
| Natural scaling?
|
| Propensity?
|
| Leaning factor?
|
| In geography for example(quoting from Wikipedia):
|
| "In physical geography and physical geology, aspect (also
| known as exposure) is the compass direction or azimuth that a
| terrain surface faces."
| gpsx wrote:
| You know what's another one from physics whose name has nothing
| to do with the actual meaning - "Gedanken experiment"
| gmfawcett wrote:
| Huh? It literally translates to "thought experiment" in
| English, which is exactly what it means.
| jffry wrote:
| I think you may have gotten whooshed by the joke - both
| "eigenvector" and "gedankenexperiment" are mashups of a
| German word and an English word
| Tainnor wrote:
| "Gedankenexperiment" is a fully German word, though, not
| a mashup.
| jxy wrote:
| Try to name the bones you used to type this sentence?
| vmilner wrote:
| 3blue1brown on this:
|
| https://m.youtube.com/watch?v=PFDu9oVAE-g&vl=en
| raydiatian wrote:
| Literally all you need.
| vmilner wrote:
| I think anyone starting a lin alg course could do a lot worse
| than watch all his "Essence of Linear Algebra" series before
| starting - then watch the relevant (c. 15 min) episodes as
| you take each lecture.
| jackconsidine wrote:
| I was learning principal component analysis a few years back
| which uses Eigenvectors to reduce feature dimensions while
| minimizing information loss (sorry if I butchered this)
|
| I was really struggling to grok what Eigenvectors and
| Eigenvalues were and found this video to be the best intuition
| primer. I wish I had 3b1b when I was in high school and college
| hackandthink wrote:
| Machine Learning (LDA):
|
| "By finding eigenvectors we'll find axes of new subspace where
| our life gets simpler: classes are more separated and data within
| classes has lower variance."
|
| https://medium.com/nerd-for-tech/linear-discriminant-analysi...
| _gmax0 wrote:
| Also, if you come from a computing background, I think
| Eigenfaces is a great, illustrative use of eigenvalues.
|
| https://en.wikipedia.org/wiki/Eigenface
| [deleted]
| Waterluvian wrote:
| Something that frustrates me, and maybe I'm just confessing my
| stupidity, is the extra layer of indirection in any discipline
| when things are named after people and not the thing's
| characteristics.
|
| My doctor once told me "if you learn enough Latin, a lot of names
| in medicine will hint at what they are, so you have less to
| memorize."
|
| I find that these names often lend a sense of complexity to
| concepts that turn out to be rather simple. In high school this
| really contributed to my struggles.
|
| Edit: apparently Eigen isn't a person's name so I sure picked an
| embarrassing moment to bring this up.
| 323 wrote:
| But you assume that there is one word to describe the
| characteristics.
|
| If such a word doesn't exist, you might as well name it after a
| person instead of trying to invent a new word.
| [deleted]
| ravi-delia wrote:
| I recently embarked on a journey to come up with a math
| vocabulary for Toki Pona, a lovely little artistic conlang
| which deserves better than what I'm doing to it. In Toki Pona,
| words are build up from simpler ones to describe a thing as it
| is. A friend is 'jan pona', a person who is good (to me, the
| speaker). So I've had to come up with names which describe math
| topics.
|
| It's awful.
|
| You know how many same-xs there are?! Eigenvalue, eigenvector,
| homomorphism, isomorphism, hom _e_ omorphism, homotopic. Which
| one gets to actually be "same shape"? Worse are when well
| meaning mathematicians use descriptive names anyway. Open and
| closed are not mutually exclusive, giving rise to the awful
| clopen (and don't pretend like ajar helps. an ajar door is an
| open door!). Groups, rings, and fields all sort of bring to
| mind the objects they describe, but only after you know the
| archetypal examples. Math is the study of giving the same name
| to different things, and that gives rise to more names than
| there are short descriptions.
|
| So do you know what I did? Whenever I could, I used a real
| person's name. It freed up a limited vocabulary, and gave
| enough wiggle room to translate most undergrad math without too
| much loss. I suspect a similar thing is in play with math.
| Maybe the category theory people have abstractions to usefully
| describe "same-functions" without confusion. But in general,
| things are named poorly because it's genuinely a hard task.
| jpmattia wrote:
| > _is the extra layer of indirection in any discipline when
| things are named after people and not the thing's
| characteristics._
|
| "Eigen" in German has same English root as "own": "Eigenvalue"
| is Germanglish for "Own/inherent value", so meets your spec of
| naming a thing after its characteristics, as long as "naming"
| is allowed to be in multiple languages.
| filmor wrote:
| It doesn't mean "same". It means "own" in the sense of
| "inherent" or "characteristic".
| jpmattia wrote:
| Fair enough, edited.
| DocTomoe wrote:
| A common language fosters research and common understanding.
|
| In IT, that language is English. In diplomacy, before
| interpreters were plentiful, that language was French. And in
| many classical, medieval-era sciences, that language was Latin
| (as a commonly-understood language that came from it's ease of
| being learned by romance-language speakers and being rather
| relevant in the (then church-run) universities).
|
| So, there's no indirection intended. It's just an artefact of
| the past - an artefact that helps Chinese, Spanish and American
| doctors communicate (in broad strokes) even today.
| constantcrying wrote:
| It is sometimes very hard to name things well. The name either
| becomes so unspecific that it is just as useless, or it gets so
| long that nobody will use it.
|
| This gets worse the "deeper" the math goes, but _for me_ it
| never was a real problem, as you usually learn the definition
| together with the name.
| cogman10 wrote:
| You see this sort of thing crop up in chemistry.
|
| For really simple compounds, names are more or less settled
| and consistent (with some exceptions).
|
| But as soon as your compound starts to get more complex
| (think organic chemistry) all the sudden, it becomes nigh
| impossible to consistently name things. There are tons of
| compounds with the same chemical formula that are regionally
| named differently. Even worse, there are tons of compounds
| with the same chemical formula that are actually different
| things due to how the compound is arranged. (Good ole carbon
| chains).
| martin_balsam wrote:
| But it is named after its characteristic, albeit in German
| Waterluvian wrote:
| Well... boy did I pick the wrong example to bring this up
| with. Alas, I'll leave my shame here for all to see.
| sfpotter wrote:
| If you learn a lot of math, a lot of names will hint at what
| they are so you have less to memorize. :-)
| Waterluvian wrote:
| Except when some smart jerk discovered like eight different
| things!
| jpmattia wrote:
| And then we have Grothendieck's prime (57), just to keep
| life interesting.
| BlueTemplar wrote:
| Mandatory not-Euler's :
|
| https://en.wikipedia.org/wiki/List_of_things_named_after_Le
| o...
| jks wrote:
| My favorite is the "Lemma that is not Burnside's". Also
| known as the orbit-counting theorem, the Polya-Burnside
| lemma, the Cauchy-Frobenius lemma, and of course Burnside's
| lemma.
| ragnese wrote:
| Or when a smart jerk discovered a thing, and then
| discovered another thing based on the first thing:
| https://en.wikipedia.org/wiki/Ramond%E2%80%93Ramond_field
| ProjectArcturis wrote:
| Who is this explainer aimed at? If you can understand the first
| sentence, you probably already know what an eigenvalue is.
| [deleted]
| 3qz wrote:
| Jyaif wrote:
| Right, but it's great to refresh your memory about eigenvalues.
| CamperBob2 wrote:
| The thing about Higham is that he's sort of a one-man Wikipedia
| of linear algebra. Many of the terms that he uses also have
| their own pages that (eventually) break the concepts down into
| comprehensible terms.
|
| See https://nhigham.com/index-of-what-is-articles/ for a useful
| listing. Or, in an alternative form,
| https://github.com/higham/what-is . Notice that if you go all
| the way back up the rabbit hole you'll find user-friendly
| articles like "What is a matrix?" that clearly define the terms
| used farther down.
|
| I really dig Higham's pedagogic style, in case it's not
| obvious.
| techwizrd wrote:
| Often, papers or terse textbooks will list a definition like
| the first sentence without the added detail below. I think this
| is great for undergraduate students or folks who'd like to
| refresh their memory a bit on eigenvalues, how they're derived,
| and what they may imply. I certainly found it helpful.
| Kalanos wrote:
| You lost us at lambda
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