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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…
I do think, the term eigenvalue is rather opaque, and should be replaced by a more plain-english terminology that readily conveys its meaning.
Can you propose one?
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."
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).
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…
"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.
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…
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.
Earlier quoted context omitted.
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…
What does 'through the origin' mean in a physical 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.
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 rewritt…
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…
> 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.
I do think, the term eigenvalue is rather opaque, and should be replaced by a more plain-english terminology that readily conveys its meaning.
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/
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