Upper triangular matrix

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero. A triangular matrix is one that is either lower triangular or upper triangular. A matrix that is both upper and lower triangular is called a diagonal matrix.

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 product of a lower triangular matrix L and an upper triangular matrix U if and only if all its leading principal minors are non-zero.

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.

Many operations on upper triangular matrices preserve the shape:

Together these facts mean that the upper triangular matrices form a subalgebra of the associative algebra of square matrices for a given size. Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the Lie algebra of square matrices of a fixed size, where the Lie bracket [a,b] given by the commutator ab-ba. The Lie algebra of all upper triangular matrices is often referred to as a Borel subalgebra of the Lie algebra of all square matrices.

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