The true fathers of quantum mechanics – Heisenberg, Born, Jordan, and later Bohr – started with matrix mechanics; it was the picture in which the classical equations of motion were easier to be understood as a limit of the new theory. That was in 1925. That was how quantum mechanics was born for the first time.
Within a year, wave mechanics was born and shown to be physically equivalent to matrix mechanics, mostly by Dirac and partly by Schrödinger. Wave mechanics instantly became popular, perhaps more popular than matrix mechanics, due to its mathematical similarity to classical field theory which is why it looked and still looks simpler to many. However, this mathematical similarity doesn't mean that the physical interpretation is the same. That's why we may blame many widespread misconceptions about quantum mechanics on the popularity of the wave mechanics.
Today, we usually use the terms "Schrödinger picture" and "Heisenberg picture" of quantum mechanics for the historical concepts wave mechanics and matrix mechanics, respectively.
It's not clear what you mean by "generalized matrix mechanics". It may mean many things, e.g. some obscure and totally inequivalent theory
with more than two indices per "operator".