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\end{bmatrix}</math>
is the lower triangular
:<math>\begin{bmatrix}
1 & 5 & 8 \\
2 & 96 & 9 \\
4 & 9 & 69
\end{bmatrix}</math>
and
:<math>\begin{bmatrix}
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\end{bmatrix}</math>
is the upper triangular
:<math>\begin{bmatrix}
1 & 4 & 1 \\
99 & 6 & 9 \\
40 & 88 & 1
\end{bmatrix}</math>
==Forward and back substitution==
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: <math>p_A(x) = (x-a_{11})(x-a_{22})\cdots(x-a_{nn})</math>,
that is, the unique degree ''n'' polynomial whose roots are the diagonal entries of ''A'' (with multiplicities).
To see this, observe that <math>xI-A</math> is also triangular and hence its determinant <math>\det(xI-A)</math> is the product of its diagonal entries <math>(x-a_{11})(x-a_{22})\cdots(x-a_{nn})</math>.<ref name="axler">{{Cite book |last = Axler | first = Sheldon Jay
==Special forms==
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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.
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.
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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.
== Algebras of triangular matrices ==
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