Editing Triangular matrix (section)

==Triangularisability{{Anchor|Triangularizability}}==
A matrix that is [[similar matrix|similar]] to a triangular matrix is referred to as '''triangularizable'''. Abstractly, this is equivalent to stabilizing a [[flag (linear algebra)|flag]]: upper triangular matrices are precisely those that preserve the [[standard flag]], which is given by the standard ordered basis <math>(e_1,\ldots,e_n)</math> and the resulting flag <math>0 < \left\langle e_1\right\rangle < \left\langle e_1,e_2\right\rangle < \cdots < \left\langle e_1,\ldots,e_n \right\rangle = K^n.</math> 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 stabilizes the standard flag.

Any complex square matrix is triangularizable.<ref name="axler"/> In fact, a matrix ''A'' over a [[field (mathematics)|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'' stabilizes a flag, and is thus triangularizable 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.<ref name="axler"/><ref name="herstein">{{Cite book | last = Herstein | first = I. N. | url = https://www.worldcat.org/oclc/3307396 | title = Topics in Algebra | date = 1975 | publisher = Wiley | isbn = 0-471-01090-1 | edition = 2nd | location = New York | oclc = 3307396 | pages = 285&ndash;290}}</ref>

In the case of complex matrices, it is possible to say more about triangularization, 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.

===Simultaneous triangularisability===
{{see also|Simultaneously diagonalizable}}
A set of matrices <math>A_1, \ldots, A_k</math> are said to be '''{{visible anchor|simultaneously triangularisable}}''' if there is a basis under which they are all upper triangular; equivalently, if they are upper triangularizable by a single similarity matrix ''P.'' Such a set of matrices is more easily understood by considering the algebra of matrices it generates, namely all polynomials in the <math>A_i,</math> denoted <math>K[A_1,\ldots,A_k].</math> Simultaneous triangularizability means that this algebra is conjugate into the Lie subalgebra of upper triangular matrices, and is equivalent to this algebra being a Lie subalgebra of a [[Borel subalgebra]].

The basic result is that (over an algebraically closed field), the [[commuting matrices]] <math>A,B</math> or more generally <math>A_1,\ldots,A_k</math> are simultaneously triangularizable. This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before. This was proven by Frobenius, starting in 1878 for a commuting pair, as discussed at [[commuting matrices]]. As for a single matrix, over the complex numbers these can be triangularized by unitary matrices.

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.{{Citation needed|reason=The existence of a common eigenvector is not clear, see https://mathoverflow.net/questions/43298/commuting-matrices-and-the-weak-nullstellensatz|date=March 2021}} In algebraic terms, these operators correspond to an [[algebra representation]] of the polynomial algebra in ''k'' variables.

This is generalized by [[Lie's theorem]], which shows that any representation of a [[solvable Lie algebra]] is simultaneously upper triangularizable, the case of commuting matrices being the [[abelian Lie algebra]] case, abelian being a fortiori solvable.

More generally and precisely, a set of matrices <math>A_1,\ldots,A_k</math> is simultaneously triangularisable if and only if the matrix <math>p(A_1,\ldots,A_k)[A_i,A_j]</math> is [[nilpotent]] for all polynomials ''p'' in ''k'' ''non''-commuting variables, where <math>[A_i,A_j]</math> is the [[commutator]]; for commuting <math>A_i</math> the commutator vanishes so this holds. This was proven by Drazin, Dungey, and Gruenberg in 1951;<ref>{{Cite journal | last1 = Drazin | first1 = M. P. | last2 = Dungey | first2 = J. W. | last3 = Gruenberg | first3 = K. W. | date = 1951 | title = Some Theorems on Commutative Matrices |url = http://jlms.oxfordjournals.org/cgi/pdf_extract/s1-26/3/221 | journal = Journal of the London Mathematical Society | language = en | volume = 26 | issue = 3 | pages = 221–228 | doi = 10.1112/jlms/s1-26.3.221}}</ref> a brief proof is given by Prasolov in 1994.<ref>{{Cite book | last = Prasolov | first = V. V. | url = https://www.worldcat.org/oclc/30076024 | title = Problems and Theorems in Linear Algebra | pages = 178–179 | date = 1994 | publisher = American Mathematical Society | others = Simeon Ivanov | isbn = 9780821802366 |location=Providence, R.I. | oclc = 30076024}}</ref> One direction is clear: if the matrices are simultaneously triangularisable, then <math>[A_i, A_j]</math> is ''strictly'' upper triangularizable (hence nilpotent), which is preserved by multiplication by any <math>A_k</math> or combination thereof – it will still have 0s on the diagonal in the triangularizing basis.