On strang’s fifth bound for eigenvalues of the jordan product of two self-adjoint operators on a finite dimensional unitary space

An almost coordinate-free proof of the algebraic version of the Mohr circles theorem is given. An alternative and somewhat simplified version of the expression for the famous fifth bound of Strang for the Jordan product of two self-adjoint operators on a finite-dimensional unitary space is found and it is proved that the associated extremum is attained in a two dimensional subspace. Details of the calculations leading to the extreme values are given.