Linear and multilinear algebra; matrix theory Definition

Upper triangular matrix

Definition

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

Remarks

Sums and products of upper triangular matrices of the same order are upper triangular. The determinant of a triangular matrix is the product of its diagonal entries. The diagonal entries of an upper triangular matrix over a field are its eigenvalues. A system $Ux = b$ with $U$ upper triangular and invertible is solved by back substitution.

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Matrix definition
Square matrix definition
Upper triangular matrix definition

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

Upper triangular matrix

Definition

Remarks

Reading path

Where this leads

Further reading

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