In this paper we present matrix exponential formulae for the geometric and spectral geometric means of positive definite matrices using a conjectured exponential formula that solved by Wasin So [Linear Algebra Appl. 379 (2004)]. Mathematics subject classification (2000): 15A24, 15A29, 15A48.

We propose a new approach to modeling time-varying relational data such as e-mail transactions based on a dynamic extension of matrix factorization. To estimate effectively the true relationships behind a sequence of noise-corrupted relational matrices, their dynamic evolutions are modeled in a space of low-rank matrices. The observed matrices are assumed as to be sampled from an exponential fa...

Analysing the properties of a probability distribution is a question of general interest. In this paper we describe the properties of the matrix-exponential class of distributions, developing some properties for the discrete case and proving the closure properties, which for the case of phase-type distributions are extended to the matrix-exponential case, this not being an immediate consequence...

Both the theoretical and practical investigations of various dynamical systems need to extend the definitions of various functions defined on the real axis to the set of matrices. To this end one uses mainly three methods which are based on 1) the Jordan canonical forms, 2) the polynomial interpolation, and 3) the Cauchy integral formula. All these methods give the same result, say g(A), when t...

The purpose of these notes is to describe how one can compute the matrix exponential e when A is not diagonalisable. This is done in Teschl by transforming A into Jordan normal form. As we will see here, it is not necessary to go this far. It su ces to transform A into block form, where each block only has one eigenvalue (up to multiplicity). For completeness we also present a proof of the Jord...

We introduce the matrix exponential as a way of modelling spatially dependent data. The matrix exponential spatial specification simplifies the loglikelihood allowing a closed form solution to the problem of maximum likelihood estimation, and greatly simplifies Bayesian estimation of the model. The matrix exponential spatial specification can produce estimates and inferences similar to those fr...