simultaneous triangularization

One basis, many operators, all triangular at once. A family of operators is simultaneously triangularizable if there is a single basis of V in which every operator in the family has upper-triangular matrix form. You triangularize them all with one change of coordinates, not one each.

Geometrically this means the whole family shares one complete invariant flag: a nested chain {0} inside V1 inside ... inside Vn = V with every Vi invariant under every operator at once. The flagpicture from the single-operator case lifts verbatim to families — finding the shared flag is finding the basis.

The headline theorem is for commuting operators: any commuting family of operators over an algebraically closed field can be simultaneously triangularized. The proof bootstraps from a common eigenvector (which commuting operators always share), passes to the quotient, and recurses. More generally, McCoy's theorem characterizes simultaneous triangularizability through polynomial conditions on commutators, and Lie's theorem extends the idea to solvable Lie algebras. Note simultaneous triangularization forces all the operators' eigenvalues onto the shared diagonal in a consistent order, but it does NOT make them simultaneously diagonalizable — that is a strictly stronger demand needing each to be individually diagonalizable as well.