I'm reading "Quantum" by Manjit Kumar, a history of quantum mechanics. It tells how Heisenberg designed an array to track the frequencies of all possible spectral lines being emitted by hydrogen electrons as they "jumped" between energy levels. Heisenberg was troubled because when he multiplied two of these matrices together, the answer was non-commutative, so that AB-BA did not always equal zero. Further on, this led to his Uncertainty Principle. I am missing his reasoning. 1. Why did he multiply two of these matrices together? 2. What made the matrix properties non-commutative? Was it because of using complex numbers inside the matrix, was it it because it was a certain type of Matrix, such as Hermitian, or none of these?