In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, two
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
and
are said to commute if
, or equivalently if their
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
is zero. A
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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 properties
* Commuting matrices preserve each other's
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s. As a consequence, commuting matrices over an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
are
simultaneously triangularizable; that is, there are
bases over which they are both
upper triangular
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal ar ...
. 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.
* Two diagonalizable matrices
and
commute (
) if they are
simultaneously diagonalizable
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique. ...
(that is, there exists an invertible matrix
such that both
and
are
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...
). Converse is valid, provided that one of the matrices has no multiple eigenvalues.
* If
and
commute, they have a common eigenvector. If
has distinct eigenvalues, and
and
commute, then
's eigenvectors are
's eigenvectors.
* If one of the matrices has the property that its minimal polynomial coincides with its
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The c ...
(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
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of two commuting
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
matrices ''A'', ''B'' with their
algebraic multiplicities (the
multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
s 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.
* Two
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
matrices commute if their
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s 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
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
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 In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if \pi: \mathfrak \to \mathfrak(V) is a finite-dimensional representation of a solvable Lie algebra, the ...
, which shows that any
representation of a
solvable Lie algebra
In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted
: mathfrak,\mathfrak/math>
that consist ...
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
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of the
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of ''n'' × ''n'' matrices under multiplication is the
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of scalar matrices.
Examples
* The identity matrix commutes with all matrices.
*
Jordan block
In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has th ...
s 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. That also means that every diagonal matrix commutes with all other diagonal matrices.
*
Circulant matrices commute. They form a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
since the sum of two circulant matrices is circulant.
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.
References
{{reflist
Matrix theory