I'll let Azzam answer with more details on why eigenvectors are not replacing M, but I just want to clarify the size of T.
T is not of size equal to M, it is a smaller than this and depends on your IB/NB settings for blocking sizes of the tile format and internal blocking of the algorithm.
So if M is made of MT by NT tiles of size MB -by - NB (in all algorithms MB=NB), descT is of size MT * IB -by- NB. By default, NB=120 and IB=20 for zheev.
In conclusion, descT is NB/IB (6 by default) times smaller tham M. That is a small difference but might be interesting. We actually have to improve that to reduce it by two.
And If you don't need eigenvectors, algorithm could be modified to destroy T on the fly and never allocate the full matrix.