Physical meaning of Eigenvalues/vectors/functions

[brtcobalt]

I understand how to calculate all three of these, but what is their physical function. How do they physically manipulate the environment we are trying to describe through mathematics? Please don't use any formulas to explain this. The formulas are easy, the physical meaning is what I'm missing.

To the best of my understanding, Eigenfunctions return a topology/manifold/etc. to it's original position. Eigenvectors expand or compress a tensor/matrix. Eigenvalues give the general rest position of a linear equation. I believe I'm misunderstanding something.

 

[HallsofIvy]

First, mathematics is not physics. Mathematical concepts are not physical concepts and mathematical object, such as vectors and linear transformations do not automatically have physical functions attached to them.

Having got that off my chest, of course we can apply mathematics to physical situations. What the mathematical concepts mean depends on the specific application.

One important area in which vectors and linear transformations are applied is in "elasticity". Suppose you take hold of a rubber sheet at two points and pull on it. How do the points on the sheet move? For constant, not too great, pulling forces, the relationship between "stress" and "strain" (the force is the "stress", the movement is the "strain") we have a linear transformation from the original postion to the final position of each point on the rubber sheet.

There will always be two directions in which all points lying on lines in those directions are just moved along the line (for the simple "stretching a rubber sheet" in one of those directions, essentially parallel to the line between the points where you are holdijg and pulling on the sheet, the points will be moved farther apart and and on the other they will be compressed to gether). The directions of those lines give the eigenvectors and the amount of "expansion" and "compression" in those directions are the (positive and negative) eigenvalues.

 

[pmsrw3]

I think of eigenvectors as the natural basis vectors of a linear system. If you start with an arbitrary coordinate system x, y, z, ... each coordinate interferes with the others. When I change x, I find that y, z, ... react. But in the eigenvector basis, each coordinate acts independently of the others. You can figure out how it will behave without worrying about anyone else. Then you just add them up to get how the whole system behaves.

 

[brtcobalt]

thank you both. I guess in the future I should say "how are Eigenstates used in physical applications". Both explanations were enlightening, but if anyone else wants to chime in then please do; there are never too many ways to understand something.

I take it that Eigenfunctions are just that, the functional of an Eigenvalue and Eigenvector. Eigenstates I also assume are the argument function.

 

[pmsrw3]

thank you both. I guess in the future I should say "how are Eigenstates used in physical applications".

Oh, well the big answer to that would have to be quantum mechanics. If you ever had a chemistry class, you probably learned about electron orbitals. Those are the eigenstates of an electron in a hydrogen atom. If you put an H atom into one, it'll stay there forever. (Actually, that's not true, because it ignores electrodynamic effects, but that's a problem with the physics, not the math.)

I take it that Eigenfunctions are just that, the functional of an Eigenvalue and Eigenvector. Eigenstates I also assume are the argument function.

I don't know just what "the functional of an Eigenvalue and Eigenvector" means. An application of eigenfunctions would be solving the diffusion equation. Suppose I have a vertical tube of clear jello, and I put a concentrated solution of dye on top. I want to know how the dye will spread in the tube. You can describe this using the eigenfunctions of the Laplacian (which are just sinusoids).

 

[Skrew]

What are you referring to?

When it's vectors, an eigenvector is a vector that when multiplied by a matrix ends up as a linear combination of itself, the eigenvalue being the scaler multiple.

 

[brtcobalt]

I'm referring to ODE, PDE, functional analysis, operator theory, and vector analysis. I could use a broader understanding of all applications, but I'm reading Methods of Mathematical Physics by Courant and Hilbert and the suffix Eigen seems to have variable definition. I guess Eigen=characteristic=equilibrium=average amount of change. Eigenvibration for instance. For a vibrating string it is described as the octave/harmonic of the initial value, which I fully understand and can conceptualize. I'm just not certain if that's accurate for all Eigenmath. It's an Eigenquandry I have.

I learn the best by discussing a subject to the point of obsession, so really any bit of information I can get from the different perspectives and interpretations out there is valuable to me. I'm really trying to pin down how Eigen can be used in so many branches of mathematics. I understand how each branch is connected, but the Eigen term tends to drift.

 

[brtcobalt]

It may just be that this is a 1955 edition and a bit dated, but I'm trying to understand the mathematics of the time and get a history of math to understand the progression in physics over time.

 

[shstar_2008]

Hi everybody
I think that each mathematical concept is born due to a need in physical applications.
In dynamics,the concepts of eigen vector and eigen value return to rotation a rigid body.in this way, a 3*3 matrix has 3 eigenvector,so there are 3 vectors in reality and for each eigen vector there is an eigenvalue.this means that for every vector there is a rotation angle.