Let $M$ be a $3 x 3$ complex matrix that is diagobalizable with eigenvalues $0$ and the other two eigenvalues on the unit circle.

Let $m_1, m_2, m_3$ be positive integers such that $gcd(m_1,m_2,m_3) = 1$ and consider the diagonal matrix $\Lambda_\theta$ with entries $e^{im_1\theta}, e^{im_2\theta}, e^{im_3\theta}$. I also know that the product $\Lambda_\theta M$ also has two eigenvalues in the unit circle for all values of $\theta$. I want to analyze the eigenvalues of the product $\Lambda_\theta M$. I know that in general it is difficult to determine the eigenvalues of the product but I want to see if the following is true:

  1. Can I choose $\theta$ so that the eigenvalues on the circle of the product are of the form $e^{i2\pi\theta_1}, e^{i2\pi\theta_2}$ where $\theta_1, \theta_2$ are linearly independent over the rational numbers.

  2. Is it possible for the eigenvalues of the product to be the same irrespective of the choice of $\theta$?


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