On using exact joint diagonalization for noniterative approximate joint diagonalization

We propose a novel, noniterative approach for the problem of nonunitary, least-squares (LS) approximate joint diagonalization (AJD) of several Hermitian target matrices. Dwelling on the fact that exact joint diagonalization (EJD) of two Hermitian matrices can almost always be easily obtained in closed form, we show how two "representative matrices" can be constructed out of the original set of all target matrices, such that their EJD would be useful in the AJD of the original set. Indeed, for the two-by-two case, we show that the EJD of the representative matrices yields the optimal AJD solution. For larger-scale cases, the EJD can provide a suboptimal AJD solution, possibly serving as a good initial guess for a subsequent iterative algorithm. Additionally, we provide an informative lower bound on the attainable LS fit, which is useful in gauging the distance of prospective solutions from optimality.