# Eigenvalues of a special product

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 $$\theta$$?