  We consider the problem of recovering a matrix from its action on a known
vector in the setting where the matrix can be represented efficiently in a
known matrix dictionary. Connections with sparse signal recovery allows for the
use of efficient reconstruction techniques such as Basis Pursuit. Of particular
interest is the dictionary of time-frequency shift matrices and its role for
channel estimation and identification in communications engineering. We present
recovery results for Basis Pursuit with the time-frequency shift dictionary and
various dictionaries of random matrices.
