The mean residual lifetime of a real-valued random variable X is the function e defined by
One of the more important properties of the mean residual lifetime function is that it determines the distribution of X. See, for example, Swartz . References to related characterizations are given by Galambos and Kotz , pages 30–35. It was established by Jupp and Mardia that this property holds also for vector-valued X. As (1·1) makes sense if X is a random symmetric matrix, it is natural to ask whether the property holds in this case also. The purpose of this note is to show that, under certain regularity conditions, the distributions of such matrices are indeed determined by their mean residual lifetimes.