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In mathematics, a '''triangular matrix''' is a special kind of [[square matrix]]. A square matrix is called '''{{visible anchor|lower triangular}}''' if all the entries ''above'' the [[main diagonal]] are zero. Similarly, a square matrix is called '''{{visible anchor|upper triangular}}''' if all the entries ''below'' the main diagonal are zero.
Because matrix equations with triangular matrices are easier to solve, they are very important in [[numerical analysis]]. By the [[LU decomposition]] algorithm, an [[invertible matrix]] may be written as the [[matrix multiplication|product]] of a lower triangular matrix ''L'' and an upper triangular matrix ''U'' [[if and only if]] all its leading principal [[minor (linear algebra)|
== Description ==
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==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.
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* The sum of two upper triangular matrices is upper triangular.
* The product of two upper triangular matrices is upper triangular.
* The [[inverse matrix|inverse]] of an upper triangular matrix, if it exists, is upper triangular.
* The product of an upper triangular matrix and a scalar is upper triangular.
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