Connection between the Hadamard and matrix products with an application to matrix-variate Birnbaum-Saunders distributions

In this paper, we establish a connection between the Hadamard product and the usual matrixmultiplication. In addition, we study some newproperties of the Hadamard product and explore the inverse problem associated with the established connection, which facilitates diverse applications. Furthermore, we propose a matrix-variate generalized Birnbaum–Saunders (GBS) distribution. Three representations of the matrix-variate GBS density are provided, one of them by using the mentioned connection. The main motivation of this article is based on the fact that the representation of the matrixvariate GBS density based on element-by-element specification does not allow matrix transformations. Consequently, some statistical procedures based on this representation, such as multivariate data analysis and statistical shape theory, cannot be performed. For this reason, the primary goal of this work is to obtain amatrix representation of thematrixvariate GBS density that is useful for some statistical applications. When the GBS density is expressed by means of a matrix representation based on the Hadamard product, such a density is defined in terms of the original matrices, as is common for many matrix-variate distributions, allowing matrix transformations to be handled in a natural way and then suitable statistical procedures to be developed. © 2011 Elsevier Inc. All rights reserved.