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# 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.

1. Are all eigenvalues of the product $AB$ are positive real numbers? If not, which property of them can we infer?

2. Is the max eigenvalue of $AB$ real? Does it has multiplicity of 1?

Thanks a lot, David