## Independence of Linear and Quadratic Transformations of Multivariate Normal Random Variables

Let X be distributed Nn(0, In) – that is, X is n-variate, standard, normally distributed. Consider the linear transformations Y = BX, where B is a k x n matrix of constants, and Z = CX, where C is an m x n matrix of constants. It follows from Theorem 5.4 that

Then Y and Z are uncorrelated and therefore independent if and only if CBT = O. More generally we have

Theorem 5.6: Let X be distributed Nn(0, In), and consider the linear transformations Y = b + BX, where b is a k x 1 vector and B a k x n matrix of constants, and Z = c + CX, where cis anm x 1 vector and C anm x n matrix of constants. Then Y and Z are independent if and only if BCT = O.

This result can be used to set forth conditions for independence of linear and quadratic transformations of standard normal random vectors:

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