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What's in a name?...The name Matrix was originally picked from a dozen or more contenders. Picked because it had an, as Captain Stearns says "internetty sound". The name is also a tip-of-the-hat to the popular 80's cartoon The Transformers. In the Transformers cartoon series, the heroes of the series, the Autobots, possessed an object named the Autobot Matrix of Leadership.
The crest of the Matrix was originally modeled after the circular patches used by the US Military forces for pilots of certain aircraft. This type of patch is popular among Naval Aviators, which is a connection to the "Carrier" nature of the Matrix.
The golden wings represent the origin of the class name and its patron, the Archangel Gabriel. The wings shielding the center sword symbolizes StarFleet's mission to both protect and defend its citizens. The raised double-edged sword also symbolizes the Matrix's ability to attack a hostile force and also defend against them.