Understanding Poisson Brackets in Symplectic Notation

[aaaa202]

Okay there is a particular equation in my book, which I just can't seem to understand intuitively. I've been staring at it for an hour now without progress, so I hope some of you can explain it.
Basically it's the one on the attached picture.
Let me introduce the notation so you can help me:
[itex]\varsigma[/itex] is a vector with the new set of canonical coordinates (Q1,...Qn,P1,...,Pn) which are viewed as function of the old coordinates [itex]\eta[/itex] = (q1,..,qn,p1,...,pn). The matrix poisson bracket [[itex]\varsigma[/itex],[itex]\varsigma[/itex]]_{[itex]\eta[/itex]} then comprise the matrix with the following poisson brackets as elements [[itex]\varsigma[/itex]l,[itex]\varsigma[/itex]k]_{[itex]\eta[/itex]}.
It should then be intuitive that this can be written as MJM^T. Where M is the jacobian matrix with elements Mij = [itex]\partial[/itex][itex]\varsigma[/itex]i/[itex]\partial[/itex][itex]\eta[/itex]j
How do I realize that?

 

[pabloenigma]

This is just the definition of Poisson brackets in symplectic notation.I don't think it follows from anywhere.
I guess,you can explicitly write down the matrices explicitly for one or two independent co-ordinates,write down the matrix J explicitly(as defined in your textbook),and we will see the matrix multiplications grinding out the non symplectic familiar poisson bracket expressions.