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Mwageringel (talk | contribs) claims involving Hilbert's Nullstellensatz are not obviously correct |
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The basic result is that (over an algebraically closed field), the [[commuting matrices]] <math>A,B</math> or more generally <math>A_1,\ldots,A_k</math> are simultaneously triangularizable. This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before. This was proven by Frobenius, starting in 1878 for a commuting pair, as discussed at [[commuting matrices]]. As for a single matrix, over the complex numbers these can be triangularized by unitary matrices.
The fact that commuting matrices have a common eigenvector can be interpreted as a result of [[Hilbert's Nullstellensatz]]: commuting matrices form a commutative algebra <math>K[A_1,\ldots,A_k]</math> over <math>K[x_1,\ldots,x_k]</math> which can be interpreted as a variety in ''k''-dimensional affine space, and the existence of a (common) eigenvalue (and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz.{{Citation needed|reason=The existence of a common eigenvector is not clear, see https://mathoverflow.net/questions/43298/commuting-matrices-and-the-weak-nullstellensatz|date=March 2021}} In algebraic terms, these operators correspond to an [[algebra representation]] of the polynomial algebra in ''k'' variables.
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.
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