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Matrix (disambiguation)for other, similarly-named pages.
A TARDIS matrix was the very sentience of a TARDIS. A matrix could be removed from a TARDIS and placed into a recepticle, which is what House did when he brought TARDISes into the bubble universe. The matrix would eventually die, and without it House could feed on the TARDIS. The Doctor's TARDIS's matrix was placed inside Idris, possibly extinguishing her own consciousness. This actually allowed him to communicate with his own ship for the first time before Idris' body finally collapsed and the matrix returned to the TARDIS. (TV: The Doctor's Wife)
Just as each TARDIS carried a mathematical duplicate of the Eye of Harmony on Gallifrey (PROSE: The Quantum Archangel), a TARDIS matrix may have served as a representation of/connection to the actual Matrix. What connection a matrix had with the heart of a TARDIS is unknown.