  We introduce almost diagonal matrices in the setting of (anisotropic)
discrete homogeneous Triebel-Lizorkin type spaces and homogeneous modulation
spaces, and it is shown that the class of almost diagonal matrices is closed
under matrix multiplication.
  We then connect the results to the continuous setting and show that the
"change of frame" matrix for a pair of time-frequency frames, with suitable
decay properties, is almost diagonal. As an application of this result, we
consider a construction of compactly supported frame expansions for homogeneous
decomposition spaces of Triebel-Lizorkin type and for the associated modulation
spaces.
