Exponential probability distribution on symmetric matrices

In this paper, we define in the most natural way a multivariate extension of the exponential distribution as a particular Wishart on the cone of positive definite symmetric matrices. We also introduce a notion of reliability function for a matrix random variable. We then show that, under a condition of invariance, the exponential distribution on symmetric matrices is characterized by a property of memoryless generalizing and including the characterization established for the real exponential distribution.