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There is a lot of good material in here, but it seems to be arranged in no particular order. The level of exposition oscillates at high speed between what is appropriate for grade school and what is appropriate for graduate school. I am going to try to straighten things out a bit. Please help! [[User:Lesnail|LeSnail]] ([[User talk:Lesnail|talk]]) 01:55, 19 March 2012 (UTC)
:I've worked a bit on the first half now. The second half is untouched. [[User:Lesnail|LeSnail]] ([[User talk:Lesnail|talk]]) 03:34, 19 March 2012 (UTC)
== False claim? ==
In the article's [https://en.wikipedia.org/w/index.php?title=Triangular_matrix&oldid=761938496#Simultaneous_triangularisability section] about simultaneous triangularisability is claimed that
{{quotation|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. In algebraic terms, these operators correspond to an [[algebra representation]] of the polynomial algebra in ''k'' variables.}}
Is the claim about common eigenvalue wrong or I'm misinterpreting it? As far I know, two commuting matrices share a common eigenvector, but not necessarily a common eigenvalue: the identity matrix ''I'' and ''2I'' share common eigenvectors, but their eigenvalues are different.
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