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== 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
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.
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