An Interesting Model with Unknown Orthogonal Design Matrix
We find such signal models in radar literature. For example, $\Gamma$ represents the transmitted signal information matrix, $\mu$ is the deterministic unknown vector representing the reflections from the target, $z_i$ is a random vector generated by the clutter surrounding the target, and $e_i$ is the receiver noise. In statistical literature, such models are called linear mixed models. You can make several assumptions, such as orthogonal signal matrix, unknown clutter covariance matrix etc., and solve a hypothesis testing problem where the goal of the detector is to detect if there is a target or not. In other words, $\mu = 0$ vs $\mu \ne 0$.