In linear algebra, two matrices and are said to commute if and equivalently, their commutator is zero. A set of matrices is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other.
If matrices and are simultaneously diagonalizable, that is, there exists a similarity matrix such that and are both diagonal, then and commute. The converse is not necessarily true, since one of the matrices could be not diagonalizable, e.g.:
If one of the matrices has the property that its minimal polynomial coincides with its characteristic polynomial, which happens in particular whenever the characteristic polynomial has only simple roots, then the other matrix can be written as a polynomial in the first.
As a direct consequence of simultaneous triangulizability, the eigenvalues of two commuting complex matricesA, B with their algebraic multiplicities can be matched up as in such a way that the multiset of eigenvalues of any polynomial in the two matrices is the multiset of the values. This theorem is due to Frobenius.
Two Hermitian matrices commute if their eigenspaces coincide. In particular, two Hermitian matrices without multiple eigenvalues commute if they share the same set of eigenvectors. This follows by considering the eigenvalue decompositions of both matrices. Let and be two Hermitian matrices. and have common eigenspaces when they can be written as and. It then follows that
The property of two matrices commuting is not transitive: A matrix may commute with both and, and still and do not commute with each other. As an example, the unit matrix commutes with all matrices, which between them do not all commute. If the set of matrices considered is restricted to Hermitian matrices without multiple eigenvalues, then commutativity is transitive, as a consequence of the characterization in terms of eigenvectors.
Lie's theorem, which shows that any representation of a solvable Lie algebra is simultaneously upper triangularizable may be viewed as a generalization.
If the product of two symmetric matrices is 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.
