Commuting matrices: Difference between revisions

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== Properties ==

Commuting matrices over an algebraically closed field are [[simultaneously triangularizable]]; indeed, overin theother complex numberswords they arewill unitarilybe simultaneouslyboth upper triangular on a same triangularizablebasis. This follows from the fact that commuting matrices preserve each others eigenspaces. If one of theboth matrices isare diagonalizable, then boththey can be simultaneously diagonalized. Moreover, if one of the matrices has only distinct eigenvalues, then the other matrix mustcan be written as a polynomial of the originalfirst.

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.