Let H be a (finite-dimensional) Hermitian matrix with algebraic numbers for its entries, all of which lie in some minimal field extension of the rational numbers; call this field ℚ(H) for short. Let's assume that the eigenvalues of H are distinct, and let D be the diagonal matrix of eigenvalues of H in non-increasing order, say. Since H is Hermitian with a non-degenerate spectrum, there is a unique unitary matrix U that diagonalizes H. The normalized eigenvectors of H comprise the columns of U. Finally, let ℚ(U,D) be the field extension of the rationals containing the matrix elements of U and D.
Is there any relationship between ℚ(H) and ℚ(U,D)? In particular, I would like to know a bound on the degree of ℚ(U,D)/ℚ(H). Also if possible, is there a way to take H (without diagonalizing it!) and calculate ℚ(U,D)? I'd also be happy with something that contains ℚ(U,D) and isn't that much bigger. (I mean, its only bigger by a factor that is constant or depends on the dimension, not on the particular H.)