Commuting matrices
Encyclopedia
In linear algebra
, a set of matrices
is said to commute if they commute pairwise, meaning 
for every pair i, j (equivalently, the commutator
vanishes:
; more abstractly, if the algebra they generate is an abelian Lie algebra).
have eigenvalues 
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 
with eigenvalues 
one can order the eigenvalues and choose the eigenbasis such that the eigenvalues of 
are 
and the eigenvalues for 
are 
This was proven by Frobenius
, with the two-matrix case proven in 1878, later generalized by him to any finite set of commuting matrices.
This is generalized by Lie's theorem, which shows that any representation of a solvable Lie algebra is simultaneously upper triangularizable, the case of commuting matrices being the abelian Lie algebra case, abelian being a fortiori solvable.
The notion of commuting matrices was introduced by Cauchy 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.
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, a set of matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
is said to commute if they commute pairwise, meaning for every pair i, j (equivalently, the commutatorCommutator
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:...
vanishes:
; more abstractly, if the algebra they generate is an abelian Lie algebra).Properties
Commuting matrices over an algebraically closed field are simultaneously triangularizable; indeed, over the complex numbers they are unitarily simultaneously triangularizable. Further, if the matrices have eigenvalues 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 with eigenvalues one can order the eigenvalues and choose the eigenbasis such that the eigenvalues of are and the eigenvalues for are This was proven by FrobeniusFrobenius
Frobenius can be* Ferdinand Georg Frobenius , mathematician** Frobenius algebra** Frobenius endomorphism** Frobenius inner product** Frobenius norm** Frobenius method** Frobenius group** Frobenius theorem...
, with the two-matrix case proven in 1878, later generalized by him to any finite set of commuting matrices.
This is generalized by Lie's theorem, which shows that any representation of a solvable Lie algebra is simultaneously upper triangularizable, the case of commuting matrices being the abelian Lie algebra case, abelian being a fortiori solvable.
History
References given in .The notion of commuting matrices was introduced by Cauchy 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.