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This paper investigates the linear precoder optimization for packet retransmissions in nonregenerative multi-input multi-output (MIMO) relay systems. To fully utilize the time diversity provided by automatic repeat request (ARQ), progressive linear precoders are designed with the objective of maximizing the mutual information delivered by multiple transmissions of the same packet. Firstly, the relay-only precoder is considered when the channel state information (CSI) is only available at the relay and the destination. We derive the form of the optimal ARQ precoder by restating the optimization problem as a matrix diagonalization problem, then the power allocation is formulated as a convex optimization problem where the Karush-Kuhn-Tucker conditions are used to obtain the optimal solution. Furthermore, we analyze the asymptotic performance when the two hops' signal-to-noise ratio (SNR) goes to infinity or zero respectively. Secondly, the joint source and relay precoder is studied assuming the CSI known at all nodes. The ARQ precoder is derived by matrix diagonalization and channel pairing between previous subchannels and current subchannels. Then the power allocation at the source and relay is formulated as a nonconvex multi-variable optimization problem where we employ an iterative algorithm to obtain the suboptimal solution. Moreover, the asymptotic performance is also analyzed with the two hops' SNR being extremely large or small, based on which the simplified method of searching channel pairing matrix is proposed. Numerical results show that the proposed precoders can improve the system performance significantly, especially in the joint source and relay precoder scenario.