Eigenvalues

[Cuddly]

How do I do them?

[Ted_Stryker]

http://en.wikipedia.org/wiki/Eigenvalue_algorithm

[Hazkali]

The answer is, you do not. Eigenvalues are of the Devil.

Well, that might be a little harsh, I'm sure its only my Linear Algebra lecturer who can make them feel like that.

Also, it would help if we knew what the application was- functions, matrices or whatever.

[averagejoe]

How do I do them?

I've never "done" eigen values. I've found them (and the associated eigen vectors, of course), used them, and know a bit about them, but I've never "done" them. I'll give a quick overview, though. Then you can ask questions.

Notation: a capitol letter is a transformation, a bold letter is a vector, a lowercase letter is a scalar. Since transformations can be represented by matricies, I will tend to use the words interchangably.

Take a transformation F. A vector v is an eigen vector of F if Fv=cv for a constant c. The constant is an eigen value.

This is sometimes confusing to people. Any number can be (and probably is) an eigen value for some transformation, so there's no special set of numbers which are "eigen values." In general, when you talk about eigen values, you're talking about the eigen values of a specific transformation or vector.

The general formula to find eigen values is:

det(F-cI)=0

then solve for c. F is a transformation, and I is the identity transformation.

What this actually means is that you turn F into matrix form (if it isn't a matrix already) then subtract c from each of the diagonal entries. (Actually, most people use the greek letter lambda instead of c, but it really doesn't matter; it just has to be any variable.) then you find the determinant of the matrix.

For example, let's say I had the matrix:

1 0 0
0 0 4
2 1 0

To find the eigen values of this matrix I would start by subtracting c times the identity matrix. I would get the matrix:

1-c 0 0
0 -c 1
2 1 -c

Then I take the determinant, as usual to get:

c^2-c^3-1+c=-c^3+c^2+c-1

then you set it equal to zero, and solve for c:

-c^3+c^2+c-1=0

-(c-1)^2(c+1)=0

c=1,-1

So the eigen values for the matrix are one and minus one. Now, if you want to find the eigen vectors it's a bit more visually complex, and I'm not sure if I can represent it here. Let me know if you have any more questions, though, and I'll answer them the best I can.

Edit: @ Hazkali: Eigen values are awsome. If you do any amount of quantum mechanics, you realize that they are so much nicer than doing messy differential equations to solve stuff. And then they keep you from having to integrate impossible functions over infinite limits. Embrace the eigen value; they are your friend.

[Cuddly]

I've never "done" eigen values. I've found them (and the associated eigen vectors, of course), used them, and know a bit about them, but I've never "done" them. I'll give a quick overview, though. Then you can ask questions.

Notation: a capitol letter is a transformation, a bold letter is a vector, a lowercase letter is a scalar. Since transformations can be represented by matricies, I will tend to use the words interchangably.

Take a transformation F. A vector v is an eigen vector of F if Fv=cv for a constant c. The constant is an eigen value.

This is sometimes confusing to people. Any number can be (and probably is) an eigen value for some transformation, so there's no special set of numbers which are "eigen values." In general, when you talk about eigen values, you're talking about the eigen values of a specific transformation or vector.

The general formula to find eigen values is:

det(F-cI)=0

then solve for c. F is a transformation, and I is the identity transformation.

What this actually means is that you turn F into matrix form (if it isn't a matrix already) then subtract c from each of the diagonal entries. (Actually, most people use the greek letter lambda instead of c, but it really doesn't matter; it just has to be any variable.) then you find the determinant of the matrix.

For example, let's say I had the matrix:

1 0 0
0 0 4
2 1 0

To find the eigen values of this matrix I would start by subtracting c times the identity matrix. I would get the matrix:

1-c 0 0
0 -c 1
2 1 -c

Then I take the determinant, as usual to get:

c^2-c^3-1+c=-c^3+c^2+c-1

then you set it equal to zero, and solve for c:

-c^3+c^2+c-1=0

-(c-1)^2(c+1)=0

c=1,-1

So the eigen values for the matrix are one and minus one. Now, if you want to find the eigen vectors it's a bit more visually complex, and I'm not sure if I can represent it here. Let me know if you have any more questions, though, and I'll answer them the best I can.

Edit: @ Hazkali: Eigen values are awsome. If you do any amount of quantum mechanics, you realize that they are so much nicer than doing messy differential equations to solve stuff. And then they keep you from having to integrate impossible functions over infinite limits. Embrace the eigen value; they are your friend.

Wow, thanks a lot. That is a way better explanation than the wiki entry. Gonna have to bookmark this.