Commuting matrices

In linear algebra, two matrices and are said to commute if , or equivalently if 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.

Characterizations and propertiesEdit

  • Commuting matrices preserve each other's eigenspaces.[1] As a consequence, commuting matrices over an algebraically closed field are simultaneously triangularizable; that is, there are bases over which they are both upper triangular. In other words, if   commute, there exists a similarity matrix   such that   is upper triangular for all  . The converse is not necessarily true, as the following counterexample shows:
     
However, if the square of the commutator of two matrices is zero, that is,  , then the converse is true.[2]
  • Two diagonalizable matrices   and   commute ( ) if and only if they are simultaneously diagonalizable (that is, there exists an invertible matrix   such that both   and   are diagonal).
  • If one of the matrices has the property that its minimal polynomial coincides with its characteristic polynomial (that is, it has the maximal degree), 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 matrices A, B with their algebraic multiplicities (the multisets of roots of their characteristic polynomials) 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.[3]
  • 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 identity 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.
  • An n × n matrix   commutes with every other n × n matrix if and only if it is a scalar matrix, that is, a matrix of the form  , where   is the n × n identity matrix and   is a scalar. In other words, the center of the group of n × n matrices under multiplication is the subgroup of scalar matrices.

ExamplesEdit

  • The identity matrix commutes with all matrices.
  • Every diagonal matrix commutes with all other diagonal matrices.[4]
  • Jordan blocks commute with upper triangular matrices that have the same value along bands.
  • If the product of two symmetric matrices is symmetric, then they must commute.
  • Circulant matrices commute. They form a commutative ring since the sum of two circulant matrices is circulant.

HistoryEdit

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.[5]

ReferencesEdit

  1. ^ Horn, Roger A.; Johnson, Charles R. (2012). Matrix Analysis. Cambridge University Press. p. 70. ISBN 9780521839402.
  2. ^ Horn, Roger A.; Johnson, Charles R. (2012). Matrix Analysis. Cambridge University Press. p. 127. ISBN 9780521839402.
  3. ^ Frobenius, G. (1877). "Ueber lineare Substitutionen und bilineare Formen". Journal für die reine und angewandte Mathematik. 84: 1–63.
  4. ^ "Do Diagonal Matrices Always Commute?". Stack Exchange. March 15, 2016. Retrieved August 4, 2018.
  5. ^ 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