Another math problem! (Not HW)

[Max™]

I started this when I was 6, college never came to mind, and I'm by definition a fringe researcher.

If you're not in string groups, or at least LQC, you're on your own. I'm essentially a modified relativist, soooo, yeah. Either the LHC is going to kill stringy models once and for all, or I'm out of luck as I've been barking up the wrong tree for a long ass time.


I'll never get paid to be a theoretical physicist, I won't even make a living doing it, maybe I could write a book or something... but I'm doing it because it keeps me up at night that we still don't have a single consistent theory which describes the Universe.


As I recall, an eigenvalue is the diagonal you get after performing a matrix calculation, generally indicating the most probable results for a given value.

[Kaylia]

As I recall, an eigenvalue is the diagonal you get after performing a matrix calculation, generally indicating the most probable results for a given value.

I know how you obtain eingein value. I also understand http://upload.wikimedia.org/math/1/e...7f061e590e.png and their relation with operator's base. However, I'm sure there is much more to vectorial analysis than this. My understanding is made of tiny bit that I found left and right, and lot of duct tape. It works for now, but it won't last me forever.

Hell, i wouldn't lie if I said that something as simple as a determinant or matrix inverse is often lost to me, because I'm using them artificially, instead of understanding the meaning behind the mathematics.

[Max™]

Mine is the top end which I've just learned how to deal with, with the bases pulled out, I've been reviewing freaking trig just for the sake of pounding it all into my head besides the shit I picked up here and there.

That matrix thing is another example, an eigenvalue is similar to a solved path integral, when everything else is canceled out, you're left with the right answer... but I'm still in the process of teaching myself how to explain it properly for those without my weird education.

[Kaylia]

Using analogy to understand something like this can only get you so far. I want to see the whole thinking and logic behind vectorial analysis. Only a good book (or classes) can help me I think, but I will have to wait till summer probably.

[Max™]

Well, that's what I'm talking about, communicating it without the math background is ridiculous.

In the equations I'm used to working with I can see what is going on. I want to be able to dismantle them and write better ones if need be.

[Sylvrdragon]

I started this when I was 6, college never came to mind, and I'm by definition a fringe researcher.

If you're not in string groups, or at least LQC, you're on your own. I'm essentially a modified relativist, soooo, yeah. Either the LHC is going to kill stringy models once and for all, or I'm out of luck as I've been barking up the wrong tree for a long ass time.


I'll never get paid to be a theoretical physicist, I won't even make a living doing it, maybe I could write a book or something... but I'm doing it because it keeps me up at night that we still don't have a single consistent theory which describes the Universe.


As I recall, an eigenvalue is the diagonal you get after performing a matrix calculation, generally indicating the most probable results for a given value.

I've clicked on a couple of your papers into Theoretical Physics, and, though I can't pretend to have understood any of the specifics, I like that what you (seem to) attack are the ASSUMPTIONS in the classic models. Sure, striking off into new directions is incredibly important for Physics, but finding and correcting inherent flaws in the foundations is far more so IMO.

The style appeals to me. I've always been better at improving/correcting existing things than developing/discovering new ones. Playing Devil's Advocate is by far my favorite pass-time (See? I'm not anti-social, overly critical, or overly analytical, I'm just trying to improve people!)

[Max™]

That's a key thing I'm trying to do, yes. The models work in certain cases, so well that they can not be discarded totally. So something needs to be adjusted. I just noticed that a certain assumption lent itself more readily to adjustment in my eyes.

I'm terribly embarrassed with my horribly rambling papers, I'm fully rewriting my theoretical model after I am comfortable enough to translate it back into a provable mathematical format so it doesn't give someone like Woozie (with a stronger mathematics background) an infarction.


That t'Hooft page really made me see the holes in my knowledge clearly, and I'll link it again because it's fantastic.

Gerard ’t Hooft, Theoretical Physics as a Challenge

[Woozie]

Out of curiosity, did you have any issues understanding pseudo-code? Object/classes concept? The actual programming language?

It was the object/classes concept. I could not understand what they meant or what they did. I eventually went into Shuemue's IRC channel and a few BG people explained it to me. After their explanation, I understood it enough that I was able to finish the programs assigned, but this was after I already failed an exam and three programming projects, so there was no way I would get better than an F in that class. So I dropped it before I got another F on my record (the first time I took the class I didn't drop it and ended up with an F).

As I recall, an eigenvalue is the diagonal you get after performing a matrix calculation, generally indicating the most probable results for a given value.

Uh, that's true, but that's a horrible way to explain it. If you multiply a matrix by the unitary matrix formed from the eigenvectors of the original matrix and on the other side by the adjoint of that unitary matrix, your result is a diagonal matrix where the diagonals are the eigenvalues.

However, you can't build the unitary matrix without knowing the eigenvectors, which you usually don't know unless you already know the eigenvalues anyways. So you could never use your definition to eigenvalues to calculate eigenvalues.

Also, not all matrices are even diagonalizable.

I know how you obtain eingein value. I also understand http://upload.wikimedia.org/math/1/e...7f061e590e.png and their relation with operator's base. However, I'm sure there is much more to vectorial analysis than this. My understanding is made of tiny bit that I found left and right, and lot of duct tape. It works for now, but it won't last me forever.

Hell, i wouldn't lie if I said that something as simple as a determinant or matrix inverse is often lost to me, because I'm using them artificially, instead of understanding the meaning behind the mathematics.

I don't remember if there's actually a meaning in the determinant or if it's just something mathematicians found out can be very useful. As long as you can calculate determinants and know how to use them when they are useful (e.g. eigenvalues and inverses etc), then your understanding of them is fine.

As for calculating eigenvalues, all it takes is a few practice problems on some basic matrices and I'm sure it will all come back to you.

Eigenvalues/vectors are really simple in matrices. It only becomes weird in quantum theory when you're finding eigenvalues/eigenvectors of functions and not matrices.

I've clicked on a couple of your papers into Theoretical Physics, and, though I can't pretend to have understood any of the specifics, I like that what you (seem to) attack are the ASSUMPTIONS in the classic models. Sure, striking off into new directions is incredibly important for Physics, but finding and correcting inherent flaws in the foundations is far more so IMO.

The style appeals to me. I've always been better at improving/correcting existing things than developing/discovering new ones. Playing Devil's Advocate is by far my favorite pass-time (See? I'm not anti-social, overly critical, or overly analytical, I'm just trying to improve people!)

The only problem I have with Max's style is that if he's not backing up what he's saying with rigorous mathematics, then he's a philosopher, not a scientist. I know he's working on the math and with his intelligence and enthusiasm, it's just a matter of time before he gets his math straight enough so that he can justify the stuff he's said. But until then, it's not really science. If you want to debunk a scientific theory or create a new one, you have to do so experimentally or mathematically. He does neither.

[joft]

I think the best way to understand eigenvalues and eigenvectors is to read about the Jordan normal form

Emma's Final Year Project
Jordan normal form - Wikipedia, the free encyclopedia (wikipedia has some advanced stuff in the article but it's only important to understand the basics)

given a linear operator T from a vector space V to itself (or equivalently a square matrix that represents that operator), there exists a basis for V in which the matrix representation of T is really nice.

to find that basis, we first try to decompose V into subspaces where each subspace is affected in a certain way by T. We look for subspaces W of V, which we call "invariant subspaces," such that T(W) is contained in W. if W has this property, T can actually be viewed as a linear operator on W (just restrict the domain). it's not hard to convince yourself that invariant subspaces correspond to eigenspaces (note: if L is an eigenvalue of A, then L has 1 or more eigenvectors, so the eigenspace corresponding to L is the subspace spanned by those eigenvectors) of a matrix representation of T (all such matrix representations are similar by changes of basis, and we're trying to find the nicest basis so that the matrix representation will be nicest).

it turns out that you can decompose the space V into a direct sum of T-invariant subspaces.

V = W1 + W2 + ... + Wr

if you know about group actions and orbits, this is a lot like the orbit decomposition of the action of a group, except in this case T may not be invertible so this is not necessarily an actual group action. since T is not invertible, there may be some "collapsing" happening, but we are still guaranteed that Wj can only collapse into a subspace of itself. so this decomposition helps understand what T does to V, or rather, it reduces the problem to finding out what T does to each Wj.

this approach is extremely important in math: whenever you encounter an object, break it apart and bust it down into "prime"/"indecomposable"/"irreducible" pieces that are easier to understand. you can easily think of dozens of examples: prime factorization of integers, cyclic decomposition of abelian groups, factorization of polynomials, "factorization" of anything really, fourier series representation of a function in terms of "simple" wave functions, etc etc etc.

the Jordan normal form J of the matrix representation of T tells you a lot of information about the invariant subspace decomposition. the diagonal entries of J are the eigenvalues of T. the number of "blocks" containing the same eigenvalue L is called the geometric multiplicity of L (i'll call it gm(L)), and it tells you the dimension of the eigenspace of L (hence the dimension of one of the invariant subspaces). the number of times L occurs on the diagonal is the algebraic multiplicity of L (which is the number of times L is a root of the characteristic polynomial), (call this am(L)).

gm(L) is always less or equal to am(L). if gm(L) is strictly less than am(L) then L will have at least one "block" with 1's above the diagonal in the matrix J. And this means there is some collapsing that happens in the corresponding invariant subspace. if am(L) = gm(L), then no collapsing occurs in that subspace. if that is true for all of the eigenvalues then T is invertible.

there's another aspect to this story, and it has to do with another important motif in mathematics: classification. If you consider any particular type of mathematical object, like a matrix, it would be nice to have a classification system which says "every matrix belongs to one class, every class has a very nice 'canonical' representative, and here is an algorithm for determining the canonical form corresponding to any given matrix." it turns out there are several different important classification schemes for matrices. First you pick an equivalence relation-- that partitions the set of matrices into equivalence classes. then you agree upon the most natural or nicest representative for each class. The Jordan normal form is what you get when your equivalence relation is matrix similarity (if B = PAQ where P and Q are invertible then B and A are similar).

So far everything I've said about eigenvalues is rather theoretical in nature, but eigenvalues are also important numerically and in applications. This link below is WONDERFUL, particularly the "Demo" videos.
MIT OpenCourseWare | Mathematics | 18.06 Linear Algebra, Spring 2005 | Tools

Finally, another matrix normal form which you might want to investigate is called the SVD, Singular-Value Decomposition. Singular values and eigenvalues/vectors are not the same thing, but (I think) they have similar geometric meanings, and even if they don't, the geometric meaning of SVD is very nice to think about for its own sake:
Singular value decomposition - Wikipedia, the free encyclopedia

[Max™]

If you want to debunk a scientific theory or create a new one, you have to do so experimentally or mathematically. He does neither.

Well, it is essentially the same as the path integral formulation of QM, with fat arrows. I've gone into more than enough detail in the LHC thread though, so I won't here.


Also: the Jordan is where I learned about eigenvalues and determinants and such, again joft displays clearly the difference between my math education and a more thorough one. Jordans and Jacobians and Hilberts and such are taken for granted in my mind, you NEED them for QM and some aspects of GR, didn't occur to me to suggest reading over them for a better understanding of the eigenvectors and such.

[Kaylia]

It was the object/classes concept. I could not understand what they meant or what they did. I eventually went into Shuemue's IRC channel and a few BG people explained it to me. After their explanation, I understood it enough that I was able to finish the programs assigned, but this was after I already failed an exam and three programming projects, so there was no way I would get better than an F in that class. So I dropped it before I got another F on my record (the first time I took the class I didn't drop it and ended up with an F).

I'm not sure if it would help or not, but think of classes as a space you define, and objects as a vector in that space. Each class contains a set of function f:X -> Y where Y can be anything (X, another object, a string, a traditional mathematical object). You could also define operations to add 2 object of this space, but it's a bit more advanced, however, it works exactly like it does in mathematics (ie: for the following class R², (x1,y1) + (x2,y2) = (x1+x2,y1+y2) could be implemented easily). Methods, operations and functions are all defined in the class itself, and there isn't any difference between them.

Class R²
  
     //Define every dimension you need to describe an element of this space.
     // It could be string of text, it could be a color, it could another object. 
     //In this case, it's 2 integer 
     int  x 
     int  y
   
     //Constructor is the default parameter to avoid a situation where 
     // (x,y) = ( , ).  Typically, you use 0, but you can also use input  
     // In most language, its name is the same as the class
     R²()        
        x = 0
        y=  0
     
    // f:R² -> R²    F set x to the constant x1, and y remain unchanged
    setX(int x1)
         x= x1          
   
    // f:R² -> R    It's pretty much a projection operator
    returnX()
         return x 

    // f:R² -> R    length
    length()
        int d 
        d=  sqrt(x²+y²)
        return d 

End Class

Thing get a bit messy in the "main" class when you interact with the user , but in overall, if you can see why a class= space, the code you write should be pretty good

[Kaylia]

Eigenvalues/vectors are really simple in matrices. It only becomes weird in quantum theory when you're finding eigenvalues/eigenvectors of functions and not matrices.

I'm getting used to use eigevalue of functions I guess. I was working on pertubation theories, which is why I came up with eigen value earlier.
I have a presentation next week about this, and I need to write a program that solve 1st order perturbation using noumerov method. It shouldn't be that hard, but I'm having trouble to discuss about that kind of thing in front of the class, when I know my comprehension is superficial. It's just...I don't know, I really want to attach these vectorial concept to something that is mathematically solid.

[Trajan]

It was the object/classes concept. I could not understand what they meant or what they did. I eventually went into Shuemue's IRC channel and a few BG people explained it to me. After their explanation, I understood it enough that I was able to finish the programs assigned, but this was after I already failed an exam and three programming projects, so there was no way I would get better than an F in that class. So I dropped it before I got another F on my record (the first time I took the class I didn't drop it and ended up with an F).



Uh, that's true, but that's a horrible way to explain it. If you multiply a matrix by the unitary matrix formed from the eigenvectors of the original matrix and on the other side by the adjoint of that unitary matrix, your result is a diagonal matrix where the diagonals are the eigenvalues.

However, you can't build the unitary matrix without knowing the eigenvectors, which you usually don't know unless you already know the eigenvalues anyways. So you could never use your definition to eigenvalues to calculate eigenvalues.

Also, not all matrices are even diagonalizable.



I don't remember if there's actually a meaning in the determinant or if it's just something mathematicians found out can be very useful. As long as you can calculate determinants and know how to use them when they are useful (e.g. eigenvalues and inverses etc), then your understanding of them is fine.

As for calculating eigenvalues, all it takes is a few practice problems on some basic matrices and I'm sure it will all come back to you.

Eigenvalues/vectors are really simple in matrices. It only becomes weird in quantum theory when you're finding eigenvalues/eigenvectors of functions and not matrices.



The only problem I have with Max's style is that if he's not backing up what he's saying with rigorous mathematics, then he's a philosopher, not a scientist. I know he's working on the math and with his intelligence and enthusiasm, it's just a matter of time before he gets his math straight enough so that he can justify the stuff he's said. But until then, it's not really science. If you want to debunk a scientific theory or create a new one, you have to do so experimentally or mathematically. He does neither.

If the determinant of a matrix is nonzero, then the matrix has an inverse, which can be calculated. If the determinant is zero then the matrix does not have an inverse, as you would be dividing by zero. Geometrically: in 2-d the determinant of a 2x2 matrix is the area of the parallelogram that gets formed from the 2 vectors. This works in larger dimensions are well, i.e. in 3-d the determinant of a 3x3 matrix is the volume of the parallelpipped that gets formed from the 3 vectors.

[Sylvrdragon]

I vote we have the title of this thread changed to "Math and Physics discussion" lol.

[Woozie]

I knew about the connection between determinant, invertability, linear independence, etc. But that just depends on whether the determinant is zero or not. It doesn't really distinguish between something with determinant 5 and determinant 30, for example. I was wondering if determinant had some sort of meaning of it's own like that.

Is there anything that the determnant tells us about the matrix? Like, if the determinant of one maxtrix is 10 and another has 30, could we say "this matrix has more of [some property] than the other matrix"? I don't know if my question makes sense.

It's kinda like how the dot product of a vector with itself tells us something specific about the vector (the square of it's length). Does the determinant of a matrix tell you anything specific about the matrix or does it just tell you what you can and cannot do with the matrix?

Edit: I had completely forgot about the geometric interpretation. I normally think of that when I think of cross products, not determinants (even though cross products are determinants, lol).

[Khamsin]

I vote we have the title of this thread changed to "Math and Physics discussion" lol.

If we did that, what would we rename the Large Hadron Collider thread?

[Demosthenes11]

reading this last page depresses me because it means I　need to go back and relearn linear algebra; I didn't understand any of it when I did it ;(

[joft]

if L is an invertible linear transformation, X is a Lebesgue measurable subset of R^n, and m denotes lebesgue measure, then

m(L(X)) = |det L|m(x)

if F is a C^1 diffeomorphism (so it's Jacobian JF has nonzero det) then for any
integrable function g

integral g dm = integral g of F |det JF| dm

[Woozie]

So, what programming languages do I need to become familiar with as a scientist and mathematician? I know for sure Java isn't one of them. I don't know why that's even required for math majors.

I'm okay at matlab, and I really suck at everything else. Which languages should I be trying to learn? I'm guessing Fortran and/or C++? Or will matlab alone be enough?

[Ramor]

I've heard from multiple people that matlab is relatively useless, and that instead most companies will just train you to use whatever particular program(s) they may use instead. But being familiar with programming in general is just a useful skill because it makes learning new languages easier if you can do at least one.

I can't say for certain though, I've heard this from other engineers I'm not sure how it'll go for you as a physicist.