# 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.

A matrix which can be brought to triangular form is called a trigonalizable matrix, cf. Trigonalizable element.

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.

#### References

 [a1] F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1959) pp. 33ff (Translated from Russian) [a2] D.M. Young, R.T. Gregory, "A survey of numerical mathematics" , 2 , Addison-Wesley (1973) pp. 921ff [a3] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, "Numerical recipes" , Cambridge Univ. Press (1986) pp. 357ff
How to Cite This Entry:
Triangular matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangular_matrix&oldid=49032
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article