The process of finding a basis in which a linear operator or matrix is represented by an upper-triangular matrix; the diagonal entries then list eigenvalues (counted with algebraic multiplicity) while entries above the diagonal encode generalized-eigenspace interactions.

Principle

Triangularization exists once the characteristic (or minimal) polynomial splits over the base field: one constructs a flag of invariant subspaces so that the operator sends each subspace into itself and has scalar action on successive quotients, producing an upper-triangular matrix by similarity.

Demonstration

Over C, Schur's theorem gives a unitary matrix U such that U* A U is upper triangular with eigenvalues on the diagonal; numerically this yields a stable reduction that reveals the spectrum without requiring full diagonalization.

Misapplication

Treating an upper-triangular matrix as if it were diagonal—assuming the diagonal alone determines the operator's conjugacy class—ignores nilpotent couplings above the diagonal and can lead to wrong conclusions about powers or exponentials.

Consequence

Triangular form makes eigenvalues explicit and facilitates computation of characteristic polynomials and iterative numerical algorithms; it is a weaker but more generally available simplification than diagonalization.

Reversal

A reversal is diagonalization: when the off-diagonal entries above the diagonal vanish in some triangular representation, the operator is diagonalizable and the triangular form collapses to a diagonal one.

Boundary

Triangularization typically requires the polynomial to split in the base field; over fields where eigenvalues lie in extensions one may need a field extension. It does not in general provide canonical blocks (unlike Jordan or rational canonical forms).

Semantic Tension

Tension appears between triangularization and Jordan form: both put eigenvalues on the diagonal, but Jordan refines the triangular structure into canonical nilpotent blocks while triangularization alone leaves noncanonical upper entries.

Synthesis

Triangularization reduces an operator to a triangular action reflecting an invariant flag: it exposes the spectrum on the diagonal and encodes generalized-eigenspace relations above the diagonal, serving as a robust intermediate simplification when full diagonalization fails or is unavailable.