The Laguerre transform, introduced by J. Keilson
and W. R. Nunn
[ibid. 5, 313-359 (1979; Zbl 0449.65086
)], J. Keilson
, W. R. Nunn
and the author [ibid. 8, 137-174 (1981; Zbl 0457.42011
)], and further studied by the author [Development of the Laguerre transform method for numerical exploration of applied probability models, Ph. D. Diss., Grad. School Management, Univ. Rochester (1981)], provides an algorithmic basis for the computation of multiple convolutions in conjunction with other algebraic and summation operations. The methods enable one to evaluate numerically a variety of results in applied probability and statistics that have been available only formally behind the ”Laplacian curtain”. For certain more complicated models, the formulation must be extended. In this paper we establish the matrix Laguerre transform, appropriate for the study of semi-Markov processes and Markov renewal processes, as an extension of the scalar Laguerre transform. The new formalism enables one to calculate matrix convolutions and other algebraic operations in matrix form. As an application, a matrix renewal function is evaluated and its limit theorem is numerically exhibited. In a recent paper by the author and Kijima (1984) the bivariate Laguerre transform has also been developed for the study of bivariate processes and distributions.