rrumpf
Full Member level 5
- Joined
- Jul 6, 2007
- Messages
- 294
- Helped
- 66
- Reputation
- 132
- Reaction score
- 43
- Trophy points
- 1,308
- Location
- El Paso, TX
- Activity points
- 3,624
The standard generalized eigen-value problem can be written as
[A][x] = v[x]
where [A] and are square matrices, [x] is an eigen-vector, and v is the eigen-value. For my problem all quantities are complex. The square matrices may be Hermitian, but I would have to think about that some more to be sure that is always the case.
The problem I have encountered is something that I think would be called a "coupled" eigen-value problem. It has the form
[A][x] + [y] = v[C][x]
[D][x] + [E][y] = v*[F][y]
where [A] through [F] are all square matrices with complex elements (again with possibly Hermitian symmetry), [x] and [y] are column vectors or eigen-vectors in some sense, and v is a constant or possibly the eigen-value. v* is the complex conjugate of v. the eigen-values have the form
v = exp(i*k*L)
where 'i' is sqrt(-1), L is a positive real constant, and k is a complex number and is the unknown here.
How do I solve this problem? I "want" to do something like this…
[ A B ][x] [C 0][x]
[ C D ][y] = v [0 F][y]
…and solve it like a standard generalized eigen-value problem, but this clearly is incorrect because I not handled v* appropriately.
I have a crude brute force solution, but it can only find real valued k's and may even miss some solutions. I need to solve this problem where k can be complex. I would like to find the complete set of eigenvalues v and eigen-vectors [x y].
Can anybody help me or point me to a solution?
Thanks!!
-Tip
[A][x] = v[x]
where [A] and are square matrices, [x] is an eigen-vector, and v is the eigen-value. For my problem all quantities are complex. The square matrices may be Hermitian, but I would have to think about that some more to be sure that is always the case.
The problem I have encountered is something that I think would be called a "coupled" eigen-value problem. It has the form
[A][x] + [y] = v[C][x]
[D][x] + [E][y] = v*[F][y]
where [A] through [F] are all square matrices with complex elements (again with possibly Hermitian symmetry), [x] and [y] are column vectors or eigen-vectors in some sense, and v is a constant or possibly the eigen-value. v* is the complex conjugate of v. the eigen-values have the form
v = exp(i*k*L)
where 'i' is sqrt(-1), L is a positive real constant, and k is a complex number and is the unknown here.
How do I solve this problem? I "want" to do something like this…
[ A B ][x] [C 0][x]
[ C D ][y] = v [0 F][y]
…and solve it like a standard generalized eigen-value problem, but this clearly is incorrect because I not handled v* appropriately.
I have a crude brute force solution, but it can only find real valued k's and may even miss some solutions. I need to solve this problem where k can be complex. I would like to find the complete set of eigenvalues v and eigen-vectors [x y].
Can anybody help me or point me to a solution?
Thanks!!
-Tip