Coupled eigenvalue-like problem

[Office_Shredder]

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

P^tAP = I the identity matrix
P^tBP = 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 P^t doesn't do anything useful unless I'm missing something big.

Basically I'm just looking for what the right starting off point is

 

[Nino MMP]

. 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!