/sci/ - Science & Math

Its like a diagonal line of lambdas or something in a matrix.

I took Linear Algebra in 1st year, and it never came up again.

I'm a physics major, and I've heard about these in at least two classes. They were never adequately explained. I guess they're just confusing mathematical shit.

>>1874336
You an engineer?

It's one of the most fundamental concepts in algebra.

OP, an eigenvalue is just a measure of how a matrix <span class="math">A[/spoiler] behaves when acting on a vector in its eigenspace <span class="math">S_\lambda[/spoiler]. In that case, it behaves just like scalar multiplication. You can then decompose any vector as a linear combination of eigenvectors of the matrix, and the action of the matrix on the vector becomes obvious.

<div class="math">A q = \lambda q \;\;\; \forall q \in S_\lambda</div>

OP here,

So I can conclude that it is just a property of a matrix? And not really "a thing"

>>1874341
>physics major
>never taken a QM class

Yeah, sure.

>>1874348
What is eigenspace then? I'm an economics major, I'm taking a second year course in mathematics now. They simply teach us how to solve problems involving for instance the eigenvalues. The theory behind it is not taught thoroughly. Which can be a bit confusing since you just do "stuff" in order to find some "stuff" which solves "stuff."

>>1874350

I'm about a month into QM right now, and yes, this is one of the classes where the professor has mentioned eigenvectors and eigenvalues. He just didn't give a very good explanation of what they really are, but I do understand most of the calculations we've been doing, so that explanation probably won't be missed. In fact, I've probably worked with eigenvectors before without even knowing it because I didn't pay attention to the terminology.

>>1874361
An eigenspace is the subspace of all eigenvectors with eigenvalue <span class="math">\lambda[/spoiler].

>>1874348
this. but in layman's terms, an eigen value is a number that (in certain cases) acts like the matrix. So, since there are a bunch of tricks to find eigen values, sometime they provide an easy means to solve a matrix equation. And, they hold alot of the properties of the matrix (or operator) and are useful in theoretical applications thusly.

>>1874349

NO!
It need not be a matrix, it is a property of an "operator".

If operator A acts on state b and gives back a*b

A(b) = a*b

then a is the eignevalue of A,
b is the eigenvector or eignefucntion of A
NOTE: NOT ALL b's will work, hence I usually need to find the specific b and a (given A), that satisfy this eq

A(b) = a*b

make sense?
Why didnt you take linear algreba? Your questions sound as dumb as an engineer!

Eigenvalue literally means "characteristic value." So it's a number that is characteristic of the original linear system of equations represented by your matrix. The set of all eigenvalues uniquely determines a system, in the sense that all systems which share common eigenvalues are in some way equivalent.

In QM, energy is quantized. If you have a vector H that represents the potential energy of a system, and a wavefunction X, then you can solve HX = EX for the eigenvalues E. Those values are the allowed energies of the system. (Exactly why the matrix form is equivalent to the wavefunction form is a little complicated.)

So in QM, if two systems have the same set of eigenvalues, then we can say they're the same thing. Two hydrogen atoms, from the point of view of physics, are equal.

You can also use the multiplicity of eigenvalues to find out the degeneracy of each energy level, which is useful when you want to find the population of each state in an environment at a particular temperature.

>>1874434
>So in QM, if two systems have the same set of eigenvalues, then we can say they're the same thing. Two hydrogen atoms, from the point of view of physics, are equal.

This isn't an entirely correct argument

Hydrogen and Antihydrogen (antiproton + positron) has the same energy eigenvalues as regular hydrogen,and no physicist will ever tell you they are the same thing.

Things get much worse when you go into the angular momentum space. The photon, gluon, z boson, w Boson, and the nucleus of deuterium are spin 1 systems, meaning they have a total angular momentum eigenvalue of 1. They are in no way identical systems.