ABSTRACT
In statistics, most interest focuses on the determinant of symmetric positive definite (or positive semi-definite) matrices, especially a variance matrix. Sometimes the determinant of the [population] variance matrix of a p-dimensional random variable x is termed the generalized variance of x. This occurs in the multivariate joint probability density function of the multivariate normal distribution and when changing variables in multiple integrals, a particular determinant, the Jacobean, is required to complete the transformation. This arises in finding the [population] mean and variance of the multivariate normal distribution. In statistical experimental design the determinant of the symmetric matrix plays a role in finding certain types of optimal designs. The chapter illustrates some of the basic properties, followed by considering partitioned matrices allowing some key properties to be proved. It also contains a surprising result which greatly simplifies many statistical calculations.
