$\complex{10}$, $P_{9}$, $M_{25}$ and $M_{52}$ are all vector spaces and each has dimension 10. By Theorem IFDVS each is isomorphic to any other.
The subspace of $M_{44}$ that contains all the symmetric matrices (Definition SYM) has dimension $10$, so this subspace is also isomorphic to each of the four vector spaces above.