This paper studies positive realness and negative imaginariness of state-space symmetric systems. First, a necessary and sufficient condition is provided for a state-space symmetric system to be, respectively, positive real and negative imaginary. Then an explicit formulation for the infimum of H-infinity norm under a mixed H-infinity and negative imaginary performance and an explicit solution for the associated optimal static output feedback control gain are developed based on symmetric negative imaginary theorem. Finally, three examples are provided to illustrate the main results of the paper. (C) 2018 Elsevier Ltd. All rights reserved.