The third moment of a random vector is a matrix which conveniently arranges all moments of order three which can be obtained from therandom vector itself. We investigate some properties of its singular value decomposition. In particular, we show that left eigenvectors corresponding to positive singular values of thethird moment are vectorized, symmetric matrices. We derive further properties under theadditional assumptions of exchangeability, reversibility and independence. Statistical applications deal with measures of multivariate skewness.
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