Summary: In this note we point out that the symmetrized real matrix is also the symmetric matrix that is the closest, in the Euclidean norm, to the matrix being symmetrized. This implies that, when symmetrizing the solutions to Riccati and Lyapunov equations, one actually replaces the solution by its closest symmetric matrix. A proof of this fact that followed a proof given by \textit{K. Fan} and \textit{A. J. Hoffman} [Proc. Am. Math. Soc. 6, 111-116 (1955; Zbl 0064.01402)] is presented. A new, calculus-based, proof is also introduced. It is shown that this result can be obtained using simple rotationals.