Triangular matrix

In mathematics, a triangular matrix is a matrix with entries that have elements either completely above (upper triangular) or completely below (lower triangular) the principal diagonal.

For example:

${\displaystyle {\begin{pmatrix}1&4&2\\0&3&4\\0&0&1\\\end{pmatrix}}}$

is upper triangular and

${\displaystyle {\begin{pmatrix}1&0&0\\2&8&0\\4&9&7\\\end{pmatrix}}}$

is lower triangular.

The product of two upper triangular matrices is upper triangular, so the set of upper triangular matrices forms an algebra. Algebras of upper triangular matrices have a natural generalisation in functional analysis which yields nest algebras.

See also: Row echelon form, LU decomposition

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