The attached is a section of the derivation of canonical transformation from the symplectic formulation. I tend to get very confused by the subscripts i and j. For me they both run from 1 to 2n and can be used interchangeably. But of course that is not the case since equation (9.53) on the attached picture specifically instructs you to transpose the matrix described by (9.51). Can someone explain what is wrong with just changing the indices in a pedagogic way - an example would be lovely too.