The multiplicity of the max eigenvalue in matrix multiplication
Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq \lambda^B_n > 0$. All eigenvalues are real; and the multiplicity of $\lambda^A_1$ and $\lambda^B_1$ are 1. I'm not sure whether the following properties are true. Could you please help me to prove or disprove them.
Thanks a lot, David