  In this paper, we study the degrees of freedom (DoF) of the symmetric
multi-relay multiple-input multiple-output (MIMO) Y channel, where three user
nodes, each with M antennas, communicate via K geographically separated relay
nodes, each with N antennas. For this model, we establish a general DoF
achievability framework based on linear precoding and post-processing methods.
The framework poses a nonlinear problem with respect to user precoders and
post-processors, as well as relay precoders. To solve this problem, we adopt an
uplink-downlink asymmetric strategy, where the user precoders are designed for
signal alignment and the user post-processors are used for interference
neutralization. With the user precoder and post-processor designs fixed as
such, the original problem then reduces to a problem of relay precoder design.
To address the solvability of the system, we propose a general method for
solving matrix equations. This method is also useful to the DoF analysis of
many other multiway relay networks. Together with the techniques of antenna
disablement and symbol extension, an achievable DoF of the symmetric
multi-relay MIMO Y channel is derived for an arbitrary setup of (K, M, N). We
show that, for K >= 2, the optimal DoF is achieved for M/N in [0,
max{sqrt(3K)/3,1}) and [(3K+sqrt(9K^2-12K))/6,infinity). We also show that the
uplink-downlink asymmetric design proposed in this paper considerably
outperforms the conventional approach based on uplink-downlink symmetry.
