Commuting matrices

In linear algebra, commuting matrices are matrices that can be multiplied together in either order without affecting the result. That is, two matrices \( A \) and \( B \) are said to commute if: \[ AB = BA \] This property is significant in many areas of mathematics and physics, particularly in quantum mechanics and functional analysis, as it relates to the simultaneous diagonalization of matrices, the representation of observables in quantum systems, and other contexts where linear transformations play a crucial role.