triangular matrix
1 Triangular Matrix
Let $n$ be a positive integer.
An upper triangular matrix^{} is of the form:
$$\left[\begin{array}{ccccc}\hfill {a}_{11}\hfill & \hfill {a}_{12}\hfill & \hfill {a}_{13}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{1n}\hfill \\ \hfill 0\hfill & \hfill {a}_{22}\hfill & \hfill {a}_{23}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{2n}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {a}_{33}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{3n}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{nn}\hfill \end{array}\right]$$ 
An upper triangular matrix is sometimes also called right triangular.
A lower triangular matrix is of the form:
$$\left[\begin{array}{ccccc}\hfill {a}_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \\ \hfill {a}_{21}\hfill & \hfill {a}_{22}\hfill & \hfill 0\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \\ \hfill {a}_{31}\hfill & \hfill {a}_{32}\hfill & \hfill {a}_{33}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {a}_{n1}\hfill & \hfill {a}_{n2}\hfill & \hfill {a}_{n3}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{nn}\hfill \end{array}\right]$$ 
A lower triangular matrix is sometimes also called left triangular.
Note that upper triangular matrices and lower triangular matrices must be square matrices^{}.
A triangular matrix is a matrix that is an upper triangular matrix or lower triangular matrix. Note that some matrices, such as the identity matrix^{}, are both upper and lower triangular. A matrix is upper and lower triangular simultaneously if and only if it is a diagonal matrix^{}.
Triangular matrices allow numerous algorithmic shortcuts in many situations. For example, if $A$ is an $n\times n$ triangular matrix, the equation $Ax=b$ can be solved for $x$ in at most ${n}^{2}$ operations.
In fact, triangular matrices are so useful that much computational linear algebra begins with factoring (or decomposing) a general matrix or matrices into triangular form. Some matrix factorization methods are the Cholesky factorization and the LUfactorization. Even including the factorization step, enough later operations are typically avoided to yield an overall time savings.
2 Properties
Triangular matrices have the following properties ( “triangular” with either “upper” or “lower” uniformly):

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The inverse^{} of a triangular matrix is a triangular matrix.

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The product of two triangular matrices is a triangular matrix.

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The determinant^{} of a triangular matrix is the product of the diagonal elements.

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The eigenvalues^{} of a triangular matrix are the diagonal elements.
The last two properties follow easily from the cofactor expansion of the triangular matrix.
Title  triangular matrix 
Canonical name  TriangularMatrix 
Date of creation  20130322 12:11:40 
Last modified on  20130322 12:11:40 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  13 
Author  Wkbj79 (1863) 
Entry type  Definition 
Classification  msc 6500 
Classification  msc 1500 
Defines  upper triangular 
Defines  lower triangular 
Defines  upper triangular matrix 
Defines  lower triangular matrix 
Defines  right triangular 
Defines  right triangular matrix 
Defines  left triangular 
Defines  left triangular matrix 