Description
A matrix of the form
is called a lower triangular matrix or left triangular matrix, and analogously a matrix of the form
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.
Read more about this topic: Triangular Matrix
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