Triangular matrix

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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.


[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
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Triangular matrix. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article