# Triangular matrix

A square matrix for which all entries below (or above) the principal diagonal are zero. In the first case the matrix is called an upper triangular matrix, in the second, a lower triangular matrix. The determinant of a triangular matrix is equal to the product of its diagonal elements.

Any $( n \times n)$- matrix $A$ of rank $r$ in which the first $r$ successive principal minors are different from zero can be written as a product of a lower triangular matrix $B$ and an upper triangular matrix $C$, [a1].
Any real matrix $A$ can be decomposed in the form $A= QR$, where $Q$ is orthogonal and $R$ is upper triangular, a so-called $QR$- decomposition, or in the form $A= QL$, with $Q$ orthogonal and $L$ lower triangular, a $QL$- decomposition or $QL$- factorization. Such decompositions play an important role in numerical algorithms, [a2], [a3] (for instance, in computing eigenvalues).
If $A$ is non-singular and $R$ is required to have its diagonal elements positive, then the $QR$- decomposition $A= RQ$ is unique, [a3], and is given by the Gram–Schmidt orthonormalization procedure, cf. Orthogonalization; Iwasawa decomposition.