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 -matrix of rank in which the first successive principal minors are different from zero can be written as a product of a lower triangular matrix and an upper triangular matrix , [a1].
Any real matrix can be decomposed in the form , where is orthogonal and is upper triangular, a so-called -decomposition, or in the form , with orthogonal and lower triangular, a -decomposition or -factorization. Such decompositions play an important role in numerical algorithms, [a2], [a3] (for instance, in computing eigenvalues).
If is non-singular and is required to have its diagonal elements positive, then the -decomposition is unique, [a3], and is given by the Gram–Schmidt orthonormalization procedure, cf. Orthogonalization; Iwasawa decomposition.
|[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|
Triangular matrix. O.A. Ivanova (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Triangular_matrix&oldid=13875