A translation matrix was a data construct that facilitated the conversion of symbols and sounds from one language to another. It was a major component of communication systems and devices such as the universal translator.
Due to its unique syntax and grammatical structure, the spoken language of the Skrreeans was difficult for the universal translator to lock onto. Therefore, it took some time to establish a translation matrix. (DS9: "Sanctuary")
The Emergency Medical Hologram has a built-in translation matrix. In 2375, The Doctor's translation matrix allowed him to understand the duotronic algorithm communication of a Druoda series 5 long-range tactical armor unit. (VOY: "Warhead")
In early 2376, the crew of the USS Voyager unsuccessfully attempted to design a translation matrix that would allow communication with the nucleogenic lifeforms attacking the ship. (VOY: "Equinox, Part II")