This letter presents a study of the correlation between the eigenvalue spectra of synaptic matrices and the dynamical properties of asymmetric neural networks with associative memories. For this type of neural network, it was found that there are essentially two different dynamical phases: the chaos phase, with almost all trajectories converging to a single chaotic attractor, and the memory phase, with almost all trajectories being attracted toward fixed-point attractors acting as memories. We found that if a neural network is designed in the chaos phase, the eigenvalue spectrum of its synaptic matrix behaves like that of a random matrix (i.e., all eigenvalues lie uniformly distributed within a circle in the complex plan), and if it is designed in the memory phase, the eigenvalue spectrum will split into two parts: one part corresponds to a random background, the other part equal in number to the memory attractors. The mechanism for these phenomena is discussed in this letter.