Linear and multilinear algebra; matrix theory Definition

Lower triangular matrix

Definition

Let $R$ be a ring and let $n \ge 0$ be an integer. A square matrix $L = (l_{ij})$ of order $n$ over $R$ is lower triangular when $l_{ij} = 0$ for all $i < j$ . It is strictly lower triangular when $l_{ij} = 0$ for all $i \le j$ .

Remarks

A matrix is lower triangular if and only if its transpose is upper triangular. Sums and products of lower triangular matrices of the same order are lower triangular. A system $Lx = b$ with $L$ lower triangular and invertible is solved by forward substitution. In an LU decomposition, the factor $L$ is lower triangular.

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Matrix definition
Square matrix definition
Lower triangular matrix definition

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Tag	`0EZSD6T9E0`
URI	`m2://0EZSD6T9E0`
Kind	definition
Writing	W1 stated
MSC	15A99
Defines	Lower triangular matrix
Uses	Square matrix

Lower triangular matrix

Definition

Remarks

Reading path

Where this leads

Further reading

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