Passenger Train Safety Act This bill addresses passenger safety issues aboard trains. Specifically, the bill directs the Federal Railroad Administration to research safety issues relating to speed signs, seat securement mechanisms, and methods for mitigating derailment injuries; requires the National Railroad Passenger Corporation (Amtrak) to address various safety issues, including developing policies for the safe use of child safety seats and implementing enhanced crew training; directs the Department of Transportation to require each railroad carrier that provides regularly scheduled intercity rail passenger or commuter rail passenger transportation to install inward- and outward-facing audio recording devices in or on all controlling locomotive cabs and cab car operating compartments on such passenger trains; prohibits entities providing regularly scheduled intercity or commuter rail passenger transportation from beginning service on certain routes unless a positive train control system is fully implemented and operational on such routes; and requires railroad carriers to periodically review and update their speed limit action plans.