View source for Triangular matrix

A square [[Matrix|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.



====Comments====
A matrix which can be brought to triangular form is called a trigonalizable matrix, cf. [[Trigonalizable element|Trigonalizable element]].

Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t0941401.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t0941402.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t0941403.png" /> in which the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t0941404.png" /> successive principal minors are different from zero can be written as a product of a lower triangular matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t0941405.png" /> and an upper triangular matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t0941406.png" />, [[#References|[a1]]].

Any real matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t0941407.png" /> can be decomposed in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t0941408.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t0941409.png" /> is orthogonal and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t09414010.png" /> is upper triangular, a so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t09414012.png" />-decomposition, or in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t09414013.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t09414014.png" /> orthogonal and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t09414015.png" /> lower triangular, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t09414017.png" />-decomposition or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t09414019.png" />-factorization. Such decompositions play an important role in numerical algorithms, [[#References|[a2]]], [[#References|[a3]]] (for instance, in computing eigenvalues).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t09414020.png" /> is non-singular and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t09414021.png" /> is required to have its diagonal elements positive, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t09414022.png" />-decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094140/t09414023.png" /> is unique, [[#References|[a3]]], and is given by the Gram–Schmidt orthonormalization procedure, cf. [[Orthogonalization|Orthogonalization]]; [[Iwasawa decomposition|Iwasawa decomposition]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,   "The theory of matrices" , '''1''' , Chelsea, reprint  (1959)  pp. 33ff  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.M. Young,   R.T. Gregory,   "A survey of numerical mathematics" , '''2''' , Addison-Wesley  (1973)  pp. 921ff</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.H. Press,   B.P. Flannery,   S.A. Teukolsky,   W.T. Vetterling,   "Numerical recipes" , Cambridge Univ. Press  (1986)  pp. 357ff</TD></TR></table>