There comes to be two main approaches to Quantum Mechanics. One approach is an equations approach which uses wavefunctions, operators and sometimes eigenstages. The other approach is a linear algebra approach that uses matrices, vectors and eigenvectors to describe quantum mechanics.
Consider an example of a quantum mechanical problem that uses linear algebra for the description of particle spin:
This allows for a more direct view of commutators as discussed in the previous article on quantum mechanics [link]. Matrices have an advantage of storing much more information elegantly and are convenient for commutations.
Matrices in fact can be written for x_hat, p_hat and other operators. Matrices are also useful for introducing more than one dimension. We can also make use of this method to give us a three-dimensional Schroedinger equation. First we will start by forming three dimensions of momentum p vectors.