Linear transformation of Gaussian Random vector

Matrix U is (PxQ) dimension gaussian random vector which is partitioned in p-vector Z and Q-vector Y. For any (PXQ) matrix A, Matrix U is transformation to U' is given by Z'=Z-AY and Y and transformation is 1:1. I need to find matrix A for which Z' and Y and statistically independent. I started with COV(Z') and COV(Y) to show that they must be diagonal with diagonal values to be variances and off-diagonal to be 0s. But I am not sure how it can help me find matrix A. Any opinions.

P.S. mean of matrix U is m_u and variance is A(u)