= \diag(\pm 1, \dots, \pm 1) $.

A proof of this theorem can be found in [@Cit:Sch68]. If the eigenvalues of $ A $ and $ B $ are distinct, then there exist $ 2^n $ different solutions with the same cost. If one or both of the matrices possess repeated eigenvalues, then the eigenvectors in the matrices $V_A$ and $V_B$ are determined only up to basis rotations, which further increases the size of the solution space. The Procrustes problem is related to a dynamical system formulation of the TSP as outlined below.

The orthogonal relaxation of the combinatorial optimization problem , given by , can be solved using a steepest descent method on the manifold of orthogonal matrices. For more details about this formulation see [@Cit:TSP_sahai]. One can pose the TSP as a constrained optimization problem of the form, $$\begin{aligned}
\label{eq:minPwithEqualityConstraints}
    \min_{P \in \mathcal{O}_n} & \; \mbox{tr} \left( A^T P^T B P \right), \\
    s.t. \; & \; G(P) = 0.\end{aligned}$$ This formulation gives rise to the following set of equations, $$\label{eq:TSP flow}
\begin{aligned}
    \dot{P}       &= -P \left( \left\{ P^T B P, A \right\} + \left \{P^T B^T P, A^T \right\} \right)
- \lambda P \left( (P \circ P)^T 