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.
Best Answer
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?