triangularizability

An operator is triangularizable if you can find a basis in which its matrix is upper-triangular — all the action lives on or above the diagonal, nothing below. This is a weaker, more achievable demand than diagonalizability, which insists everything off the diagonal vanish.

The clean criterion: T is triangularizable over the field exactly when its characteristic polynomial splits into linear factors over that field. Over the complex numbers every polynomial splits, so every complex operator is triangularizable — that is Schur's theorem without the unitary refinement. Over the reals it can fail: a rotation by 90 degrees has characteristic polynomial x^2 + 1, which does not split, and indeed it fixes no line, so no triangular form exists.

Triangular form is enough for a surprising amount. The diagonal entries are the eigenvalues with algebraic multiplicity, so you read off determinant (their product) and trace (their sum) instantly, and powers and the matrix exponential become tractable. What it does not give you, that diagonalization would, is independence of the blocks — the strictly-upper part still couples the coordinates.