On the Independence of Linear and Quadratic Forms in Matrix Normal Distribution and Wishart Distribution

Accepted - June 2024

Abstract

It is well-known that the Craig-Sakamoto theorem establishes the independence of two linear forms and two quadratic forms in normal variates. Replacing the random normal vectors by the random normal matrices and Wishart variates, in this paper, we investigate interconnections between the independence of linear forms, quadratic forms, trace forms in matrix normal distribution and Wishart distribution. We show that the Craig-Sakamoto theorem still establishes the independence of both two linear forms and two quadratic forms in matrix normal distribution, but it fails establishing the independence of two linear forms and two quadratic forms in Wishart variates.