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
So… fixed points?
What is an eigenvalue?
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Re: What is an eigenvalue?
#152Eigen-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
So… fixed points? https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
Yes except eigenvalues need not be 1. e^(kx) is an eigenvector of d/dx for all k but is only a fixed point when k=1
Re: What is an eigenvalue?
#153Earlier quoted context omitted.
So… fixed points? https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
Yes except eigenvalues need not be 1. e^(kx) is an eigenvector of d/dx for all k but is only a fixed point when k=1
Right, in topology a “point” can represent any mathematical object e.g 1 of “e^(kx)”
Re: What is an eigenvalue?
#154Earlier quoted context omitted.
> 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.
It doesn't mean "same". It means "own" in the sense of "inherent" or "characteristic".
How I learnt/taught myself when I first studied this subject, was characteristic or character. People change when they come into money, or rather their character changes (in my head I called them characterless). People with character don't change. Characteristic vector/Eigen vector is something which doesn't change.
Re: What is an eigenvalue?
#155Earlier quoted context omitted.
There's not really a connection here, the category of vector spaces isn't cartesian closed.
The category of Chu spaces is through.
No, it isn't. A star-autonomous cartesian category is just a preorder. Chu(Set, n) is star-autonomous but not a preorder, and therefore not cartesian. Or more concretely: cartesian categories are models for type systems with copying and deleting, but the Chu construction builds a model for a linear type system.
Re: What is an eigenvalue?
#156By far the best explanation of eigenvalues is found on 3blue1brown: https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQ... Seriously, the course on the essence of linear algebra did more for me than my entire college linear algebra course. It's only a pity I discovered the videos at the end of my mechanical engineering degree and not sooner.
Steve Brunton and Nathan Kutz have great content for applied math also
https://www.youtube.com/watch?v=QYS-ML_vn4k&list=PLMrJAkhIeN...
Re: What is an eigenvalue?
#157Earlier quoted context omitted.
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
5z + 2 is linear?
This is confusing terminology. Functions of the form ax+b are often called “linear”, but they're only linear in the general sense if b=0.
Re: What is an eigenvalue?
#158Eigen-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
Functions satisfy the laws of a vector space (well, certain classes of functions, such as L^2 anyway -- I'm certainly no expert here). I.e., functions are vectors. And the derivative is a linear operator on that space: d/dx(a f + b g) = a d/dx(f) + b d/dx(g). Therefore d/dx has eigenvalues, one of which you properly identified as e^x (and more generally, e^ax, as others have pointed out).
> (well, certain classes of functions, such as L^2 anyway -- I'm certainly no expert here)
For any vector space V over a field F, the set of functions V -> V is a vector space over F. This is essentially because the vector space properties are carried over "point-wise".
When you consider only certain classes of functions (such as L^2), the question of whether they form a vector space boils down to the question whether those classes are closed under addition and scalar multiplication. All other vector space properties are satisfied because they are in the larger vector space of all functions.
Re: What is an eigenvalue?
#159Eigen-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
So… fixed points? https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
> So… fixed points?
Fixed directions. The operator operating on an eigenvector doesn't change the vector's direction, but can change its magnitude (i.e. length). (Except, if you change the length from positive to negative, then you kind of change, or flip, the direction to 180 degrees opposite.)
Re: What is an eigenvalue?
#160Earlier quoted context omitted.
affine, not linear. describes a line that doesn't go through the origin. that pesky shift breaks linearity 5(2z) + 2 != 2(5z + 2)
Good point. But what makes studying these functions interesting? Like what key theorems govern this class of functions?
All of linear algebra is based on linear functions. Of course that doesn't mean that you can't study affine functions with it, but it adds a layer of extra complication.
For example, linear functions over finite dimensional vector spaces can be represented with matrices which means that everything you can compute about matrices you can also compute about linear functions.