Consider the correlation matrix . We can perform an eigenvector decomposition so that = SAS T , where A is a diagonal matrix of

Consider the correlation matrix Ω. We can perform an eigenvector decomposition so that Ω = SAS^T, where A is a diagonal matrix of eigenvalues and S is the eigenvector matrix whose determinant is 1. Suppose now that z is a vector of independent and identically distributed N(0,1) random variates. Show that x = S√Dz is a vector whose correlation structure corresponds to Ω.