Heisenberg's reasoning concerning matrices

[jackmann]

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?

 

[Valerie Park]

3. How did this lead to the development of the Uncertainty Principle? 1. Heisenberg multiplied two of these matrices together to study the behavior of electrons as they transition between energy levels. This allowed him to analyze the relationship between the energies of different states and their corresponding frequencies. 2. The matrix properties were non-commutative because of Heisenberg's use of complex numbers inside the matrix. This caused the matrix elements to be dependent on both their position in the matrix and their values, which led to non-commutativity. 3. Heisenberg's discovery of the non-commutativity of the matrices led him to develop the Uncertainty Principle. He realized that it was impossible to know both the exact position and momentum of a particle at the same time due to the inherent uncertainty in the matrices. This led to the concept of indeterminacy and the Heisenberg Uncertainty Principle.