Triangular Matrix - Triangularisability

Triangularisability

A matrix that is similar to a triangular matrix is referred to as triangularisable. Abstractly, this is equivalent to stabilising a flag: upper triangular matrices are precisely those that preserve the standard flag, which is given by the standard ordered basis and the resulting flag All flags are conjugate (as the general linear group acts transitively on bases), so any matrix that stabilises a flag is similar to one that stabilises the standard flag.

Any complex square matrix is triangularisable. In fact, a matrix A over a field containing all of the eigenvalues of A (for example, any matrix over an algebraically closed field) is similar to a triangular matrix. This can be proven by using induction on the fact that A has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that A stabilises a flag, and is thus triangularisable with respect to a basis for that flag.

A more precise statement is given by the Jordan normal form theorem, which states that in this situation, A is similar to an upper triangular matrix of a very particular form. The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.

In the case of complex matrices, it is possible to say more about triangularisation, namely, that any square matrix A has a Schur decomposition. This means that A is unitarily equivalent (i.e. similar, using a unitary matrix as change of basis) to an upper triangular matrix; this follows by taking an Hermitian basis for the flag.