If there are N degree of freedom, then 2N canonical transformation, we care if a transformation realized by matrix m, is PB preserved?

Use a one-degree e.o.m example

we borrow a matrix J off diagonal to express the intertwined or simplectic form of e.o.m by Hamilton.

then more degree of freedom:

So referring back to the m matrix that do the coordinates transformation that we want it to be canonical,

Why it’s useful, we can solve complex problems like below

Note MJM^T need to discern from MJM^-1, the former is explained below conditioned for simplectic matrix, while the latter is more common, leading to similar matrix to J, in the case of unitary rotation, M^-1 = M^T, then the two takes exact form MJM^T though.