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.
What is an eigenvalue?
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Re: What is an eigenvalue?
#823blue1brown on this: https://m.youtube.com/watch?v=PFDu9oVAE-g&vl=en
Literally all you need.
Re: What is an eigenvalue?
#83I do think, the term eigenvalue is rather opaque, and should be replaced by a more plain-english terminology that readily conveys its meaning.
You know what's another one from physics whose name has nothing to do with the actual meaning - "Gedanken experiment"
Huh? It literally translates to "thought experiment" in English, which is exactly what it means.
Re: What is an eigenvalue?
#84I 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 rewritt…
Re: What is an eigenvalue?
#85Earlier quoted context omitted.
Natural frequencies of mechanical systems are eigenvalues of it’s equation of motion.
Any references to help me unpack what you just said there?
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.
Re: What is an eigenvalue?
#863blue1brown on this: https://m.youtube.com/watch?v=PFDu9oVAE-g&vl=en
Literally all you need.
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.
Re: What is an eigenvalue?
#87I 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 rewritt…
In fact functions are just infinite-dimensional vectors. Almost all of the theory goes through unchanged. This is the basic idea of functional analysis.
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.
Re: What is an eigenvalue?
#88I 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 rewritt…
In fact functions are just infinite-dimensional vectors. Almost all of the theory goes through unchanged. This is the basic idea of functional analysis.
Or rather size-of-their-domain-dimensional?
Re: What is an eigenvalue?
#89I 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 rewritt…
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.
Re: What is an eigenvalue?
#90Earlier quoted context omitted.
Any references to help me unpack what you just said there?
Landau/Lifshitz: "Mechanics" has a chapter on small oscillations
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"