We obtain a classification of Borel measurable, GL(n) covariant, symmetric-matrix-valued valuations on the space of n-dimensional convex polytopes. The only ones turn out to be the moment matrix corresponding to the classical Legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by E. Lutwak, D. Yang, and G. Zhang.
"Ellipsoids and matrix-valued valuations." Duke Math. J. 119 (1) 159 - 188, 15 July 2003. https://doi.org/10.1215/S0012-7094-03-11915-8