Simulations confirm my claim, but I can't prove it. Please help me out!

The problem in multivariate case is a half page long pdf file. The univariate case is as follows.

For given (real valued) scalars, (a11,..a22), (real-valued) x11,..,x22 must satisfy the following restrictions.

a11*x11+a12*x12=1,

a12*x12+a22*x22=1,

Let S_A=[a11^2 a12^2;a21^2 a22^2] and

S_X=[x11^2 x21^2;x12^2 x22^2]

Let rho(B)=maximum eigenvalue of B in absolute value.

My claim is that

the maximum eigenvalue of rho(S_A)<1 if and only if rho(S_X)>1 for S_X with x11,..x22 are subject to the constraints above.