TITLE: Constructing a unitary matrix
QUESTION [8 upvotes]: Setting:
Given a set of $n\times n$ matrices $A_i$, I would like to find a linear combination of these matrices $Q = \sum_i A_i x_i$ with $x_i$ a set of complex numbers, such that $Q$ is unitary: $Q^{\dagger} Q = 1$. This problem is equivalent to solving a system of quadratic equations over real numbers. As far as I understand, there is no general and efficient way to solve such systems of equations, and black-box algorithms, such as Gröbner basis, struggle with systems of around 10 variables.
Question:
Does the particular structure of this system of equations make it easier than a generic one, and can this be utilized in order to speed up the calculation.
Motivation:
This problem arises in many contexts. For example, it is related to search of symmetry of a quantum Hamiltonian. Hamiltonian $H$, a finite Hermitian matrix has a symmetry if it commutes with some unitary matrix $U$ other than identity. It is very easy to construct the linear space to which $U$ should belong, it is given by the kernel of the system of linear equations $H A - A H = 0$. However the next step requires verifying  whether this linear space contains unitary matrices other than identity.
Secondary question:
Given that it seems unlikely that there is an easy answer to the main question, I would also like to ask whether there are known classes of systems of quadratic equations that are quickly solvable numerically.

REPLY [2 votes]: Every Hermitian matrix (in fact, every normal matrix) commutes with infinitely many unitary matrices:
Lemma: The square matrices A and B commute if they can be simultaneously diagonalized.
Proof: Let A=Q D inv(Q) and B=Q E inv(Q), where D and E are diagonal. Then AB = Q D E inv(Q) = Q E D inv(Q) = B A, since diagonal matrices commute.
Corollary: If H is diagonalized by the unitary matrix Q, then U = Q D Q' is unitary for any diagonal matrix D whose entries lie on the unit circle, and U commutes with H. 
Thus, once you have the eigenvectors of your (discretized) Hamiltonian, you can easily form an infinite number of unitary matrices that commute with it. 
Is there a constraint on the symmetrices that I'm missing?