Triangular Matrix - Description

Description

A matrix of the form

L= \begin{bmatrix} l_{1,1} & & & & 0 \\ l_{2,1} & l_{2,2} & & & \\ l_{3,1} & l_{3,2} & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ l_{n,1} & l_{n,2} & \ldots & l_{n,n-1} & l_{n,n} \end{bmatrix}

is called a lower triangular matrix or left triangular matrix, and analogously a matrix of the form

U = \begin{bmatrix} u_{1,1} & u_{1,2} & u_{1,3} & \ldots & u_{1,n} \\ & u_{2,2} & u_{2,3} & \ldots & u_{2,n} \\ & & \ddots & \ddots & \vdots \\ & & & \ddots & u_{n-1,n}\\ 0 & & & & u_{n,n} \end{bmatrix}

is called an upper triangular matrix or right triangular matrix. The variable L (standing for lower or left) is commonly used to represent a lower triangular matrix, while the variable U (standing for upper) or R (standing for right) is commonly used for upper triangular matrix. A matrix that is both upper and lower triangular is diagonal.

Matrices that are similar to triangular matrices are called triangularisable.

The standard operations on triangular matrices preserve the triangular shape:

  • The sum of two upper triangular matrices is upper triangular.
  • The product of two upper triangular matrices is upper triangular.
  • The inverse of an invertible upper triangular matrix is upper triangular.
  • The product of an upper triangular matrix by a constant is an upper triangular matrix.

Together these facts mean that the upper triangular matrices form a Lie subalgebra of the Lie algebra of square matrices for any given size. The Lie algebra of all upper triangular matrices is often referred to as the Borel subalgebra, denoted . The analogous results hold for lower triangular matrices, so they also form a Lie subalgebra. However, note that the product of a lower triangular with an upper triangular matrix is not necessarily triangular.

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