Exact cospectrality probabilities for uniform random matrices

We study the conjugation action of orthogonal matrices on symmetric random matrices. Given a fixed orthogonal matrix over an algebraic number field and a random matrix with entries sufficiently uniform in the ring of integers, we wonder what the probability is that the conjugate is again integral. Our main result establishes an exact formula for this probability in terms of the Smith ideals associated to the orthogonal matrix. As an illustrative application, we establish exact formulas for the expected number of rational orthogonal matrices that preserve the integrality of a random matrix for every fixed denominator in dimensions two and three. Notably, the dependence on the denominator turns out to be non-monotone due to number-theoretic fluctuations. We also prove bounds on the probability of rational cospectrality with bounded but arbitrarily large denominator.