Commuting matrices

In linear algebra, two matrices ${\displaystyle A}$ and ${\displaystyle B}$ are said to commute if ${\displaystyle AB=BA}$ and equivalently, their commutator ${\displaystyle [A,B]=AB-BA}$ is zero. A set of matrices ${\displaystyle A_{1},\ldots ,A_{k}}$ 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, in other words they will be both upper triangular on a same basis. This follows from the fact that commuting matrices preserve each others eigenspaces. If both matrices are diagonalizable, then they can be simultaneously diagonalized. Moreover, if one of the matrices has only distinct eigenvalues, then the other matrix can be written as a polynomial of the first.

Further, if the matrices ${\displaystyle A_{i}}$  have eigenvalues ${\displaystyle \alpha _{i,m},}$  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 ${\displaystyle A,B}$  with eigenvalues ${\displaystyle \alpha _{i},\beta _{j},}$  one can order the eigenvalues and choose the eigenbasis such that the eigenvalues of ${\displaystyle A+B}$  are ${\displaystyle \alpha _{i}+\beta _{i}}$  and the eigenvalues for ${\displaystyle AB}$  are ${\displaystyle \alpha _{i}\beta _{i}.}$  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.

Examples

• Diagonal matrices commute
• Jordan blocks commute with upper triangular matrices that have the same value along bands.
• If the product of two symmetric matrices are symmetric, then they must commute

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.^[1]

References

1. Drazin, M. (1951), "Some Generalizations of Matrix Commutativity", Proceedings of the London Mathematical Society, 3, 1 (1): 222–231, doi:10.1112/plms/s3-1.1.222