Commuting matrices: Difference between revisions

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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>.

Further, if the matrices <math>A_i</math> have eigenvalues <math>\alpha_{i,m},</math> 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 <math>A,B</math> with eigenvalues <math>\alpha_i, \beta_j,</math> one can order the eigenvalues and choose the eigenbasis such that the eigenvalues of <math>A+B</math> are <math>\alpha_i + \beta_i</math> and the eigenvalues for <math>AB</math> are <math>\alpha_i\beta_i.</math> This was proven by [[Ferdinand Georg Frobenius|Frobenius]], with the two-matrix case proven in 1878, later generalized by him to any finite set of commuting matrices. Another proof, using [[Hilbert's Nullstellensatz]] is sketched in the article of this name.

[[Lie's theorem]], which shows that any representation of a [[solvable Lie algebra]] is simultaneously upper triangularizable may be viewed as a generalization.