My Claim below, if proved, would be extremely useful (in economics).
Simulations seem to confirm my claim, but I can't prove it. Please help me!
For given (real valued) scalars, (a11,..a22), (real-valued) x11,..,x22 must satisfy the following restrictions.
Define the following matrices.
S_A=[a11^2 a12^2;a21^2 a22^2]
S_X=[x11^2 x21^2;x12^2 x22^2] : ((2,1)-th element is x12^2, not x21^2)
Let rho(B)=maximum eigenvalue of n by n matrix, B in absolute value.
My Claim : The following assertions are equivalent:
(2) rho(S_X)>1 for any S_X with x11,..x22 obeying the constraints above.
My Claim in general setup is given in the half-page long attached pdf file.