On a minimization problem associated with linear dynamical systems (Q1184484)

The author solves the following minimization problem: let \(A\) and \(B\) be arbitrary fixed symmetric linear transformations in an orthonormal basis of real \(n\)-dimensional Euclidean vector space; find such symmetric commuting matrices \(P\), \(Q\) \((PQ=QP)\) for which the distance between \((A,B)\) and \((P,Q)\) is minimal in the set of all symmetric commuting pairs of matrices. This problem has many applications in the theory of linear dynamical systems of the second order. The author considers the case \(n=2\) in detail (the case of linear dynamical systems with two degrees of freedom).