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In [[linear algebra]], two [[matrix (mathematics)|matrices]] <math>A</math> and <math>B</math> are said to '''commute''' if <math>AB=BA</math> and equivalently, their [[commutator]] <math>[A,B]= AB-BA</math> is zero. A set of matrices <math>A_1,\ldots,A_k</math> 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 the property that its minimal polynomial coincides with its characteristic polynomial (i.e., 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 of the first. | |||
As a direct consequence of simultaneous triangulizability, the eigenvalues of two commuting matrices complex ''A'', ''B'' with their algebraic multiplicities (the [[multiset]]s of roots of their characteristic polynomials) can be matched up as <math>\alpha_i\leftrightarrow\beta_i</math> in such a way that the multiset of eigenvalues of any polynomial <math>P(A,B)</math> in the two matrices is the multiset of the values <math>P(\alpha_i,\beta_i)</math>. | |||
[[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 is symmetric, then they must commute. | |||
== History == | |||
The notion of commuting matrices was introduced by [[Arthur Cayley|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 [[Ferdinand Georg Frobenius|Frobenius]] in 1878.<ref>{{Citation | first = M. | last = Drazin | journal = Proceedings of the London Mathematical Society | year = 1951 | series = 3 | volume = 1 | pages = 222–231 | doi=10.1112/plms/s3-1.1.222 | issue = 1 | title = Some Generalizations of Matrix Commutativity}}</ref> | |||
== References == | |||
{{reflist}} | |||
[[Category:Matrix theory]] | |||
Latest revision as of 15:16, 26 August 2013
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.
![{\displaystyle [A,B]=AB-BA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a3b93b316dd0b6b0ab2c71e486c901ddfe6e79a)
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 the property that its minimal polynomial coincides with its characteristic polynomial (i.e., 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 of the first.
As a direct consequence of simultaneous triangulizability, the eigenvalues of two commuting matrices complex 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 .
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 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.[1]
References
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