Abstract
We study the problem of minimizing a quadratic quantity defined for given two Hermitian matrices X, Y and a positive-definite Hermitian matrix. This problem is reduced to the simultaneous diagonalization X, Y when XY=YX. We derive a lower bound for the quantity, and in some special cases solve the problem by showing that the lower bound is achievable. This problem is closely related to a simultaneous measurement of quantum mechanical observables which are not commuting and has an application in the theory of quantum state estimation.
