Commuting matrices

In linear algebra, a set of matrices $A_{1},\ldots ,A_{k}$ is said to commute if they commute pairwise, meaning $A_{i}A_{j}=A_{j}A_{i}$ (equivalently, the commutator vanishes: $[A_{i},A_{j}]$).

Properties

Commuting matrices over an algebraically closed field are simultaneously triangulizable; indeed, over the complex numbers they are unitarily simultaneously triangulizable. 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.

History

References given in (Drazin 1951).

The notion of commuting matrices was introduced by Cauchy 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.

References

• Drazin, M. (1951), Proceedings of the London Mathematical Society, 3, 1 (1): 222–231, doi:doi:10.1112/plms/s3-1.1.222 {{citation}}: Check |doi= value (help); Missing or empty |title= (help)