Coupled eigenvalue-like problem

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In summary, the problem states that for complex symmetric matrices A and B and their determinant having three distinct solutions a, b and c, there exists an invertible matrix P such that P transpose times A times P equals the identity matrix and P transpose times B times P equals a diagonal matrix with non-zero entries a, b, and c. The solution involves constructing three distinct vectors x, y, and z such that aAx = Bx, bAy = By, and cAz = Bz. However, a, b, and c must be distinct, and using P = [x y z] may not be sufficient. The problem can be solved without using eigenvalues.
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Homework Statement


If A, B are (complex) symmetric 3x3 matrices and det(xA-B) has three distinct solutions a,b and c, prove there exists P invertible such that

PtAP = I the identity matrix
PtBP = diag(a,b,c) the diagonal matrix with non-zero entries a,b and c

Homework Equations



det(xA-B)=0 iff there exists v =/= 0 s.t. xAv = Bv

The Attempt at a Solution



So we have three distinct vectors we'll call x,y and z such that

aAx = Bx
bAy = By
cAz = Bz

and from here we want to construct P. The first thing I notice is that Ax, Ay and Az are all non-zero, as otherwise Bx (or y or z) is too and one of a,b or c can be arbitrary (and hence they are not necessarily distinct). But then just because a, b and c aren't necessarily distinct doesn't mean that there aren't distinct a, b and c that do satisfy the condition. This way lies madness, so I decided to go another route:

If we naively try P = [x y z] then AP = [Ax Ay Az] and BP = [aAx bAy cAz] which looks like a pretty good start, but then Pt doesn't do anything useful unless I'm missing something big.

Basically I'm just looking for what the right starting off point is
 
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. I feel like this problem should be doable without the use of eigenvalues, but I'm not sure how to go about it. Any help is appreciated!
 

Related to Coupled eigenvalue-like problem

1. What is a coupled eigenvalue-like problem?

A coupled eigenvalue-like problem is a mathematical problem that involves finding eigenvalues and eigenvectors for a system of equations that are interconnected or "coupled" to each other. This means that the equations cannot be solved independently, and the eigenvalues and eigenvectors must be found simultaneously.

2. What is the significance of solving a coupled eigenvalue-like problem?

Solving a coupled eigenvalue-like problem can provide important insights into the behavior and properties of complex systems, such as in quantum mechanics or fluid dynamics. It can also help in predicting the stability and dynamics of a system.

3. How is a coupled eigenvalue-like problem solved?

There are various numerical methods for solving a coupled eigenvalue-like problem, such as the power method, Jacobi method, or QR algorithm. These methods involve iteratively finding the eigenvalues and eigenvectors of the system until a desired level of accuracy is achieved.

4. What are some real-world applications of coupled eigenvalue-like problems?

Coupled eigenvalue-like problems have numerous applications in physics, engineering, and other fields. For example, they can be used to study the vibrations of structures, the behavior of electric circuits, or the properties of molecules in chemistry.

5. What are the differences between a coupled eigenvalue-like problem and a traditional eigenvalue problem?

In a traditional eigenvalue problem, the equations are independent and can be solved separately. In contrast, a coupled eigenvalue-like problem involves interconnected equations that must be solved simultaneously. This makes the problem more complex and often requires the use of specialized numerical methods.

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