Commuting matrices

In linear algebra, two matrices $A$ and $B$ are said to commute if $AB=BA$ and equivalently, their commutator $[A,B]=AB-BA$ is zero. A set of matrices $A_{1},\ldots ,A_{k}$ is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other.

Properties

Commuting matrices over an algebraically closed field are simultaneously triangularizable; indeed, over the complex numbers they are unitarily simultaneously triangularizable. Further, if the matrices $A_{i}$ have eigenvalues $\alpha _{i,m},$ then a simultaneous eigenbasis can be chosen so that the eigenvalues of a polynomial in the commuting matrices is the polynomial in the eigenvalues. For example, for two commuting matrices $A,B$ with eigenvalues $\alpha _{i},\beta _{j},$ one can order the eigenvalues and choose the eigenbasis such that the eigenvalues of $A+B$ are $\alpha _{i}+\beta _{i}$ and the eigenvalues for $AB$ are $\alpha _{i}\beta _{i}.$ This was proven by Frobenius, with the two-matrix case proven in 1878, later generalized by him to any finite set of commuting matrices. Another proof, using Hilbert's Nullstellensatz is sketched in the article of this name.

Lie's theorem, which shows that any representation of a solvable Lie algebra is simultaneously upper triangularizable may be viewed as a generalization.

History

The notion of commuting matrices was introduced by Cayley in his Memoir on the theory of matrices, which also provided the first axiomatization of matrices. The first significant results proved on them was the above result of Frobenius in 1878.