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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.
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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====
 
====Comments====
 
A matrix which can be brought to triangular form is called a trigonalizable matrix, cf. [[Trigonalizable element|Trigonalizable element]].
 
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]]].
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Any $  ( n \times n) $-
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matrix $  A $
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of rank $  r $
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in which the first $  r $
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successive principal minors are different from zero can be written as a product of a lower triangular matrix $  B $
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and an upper triangular matrix $  C $,  
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[[#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).
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Any real matrix $  A $
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can be decomposed in the form $  A= QR $,  
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where $  Q $
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is orthogonal and $  R $
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is upper triangular, a so-called $  QR $-
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decomposition, or in the form $  A= QL $,  
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with $  Q $
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orthogonal and $  L $
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lower triangular, a $  QL $-
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decomposition or $  QL $-
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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]].
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If $  A $
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is non-singular and $  R $
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is required to have its diagonal elements positive, then the $  QR $-
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decomposition $  A= RQ $
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is unique, [[#References|[a3]]], and is given by the Gram–Schmidt orthonormalization procedure, cf. [[Orthogonalization|Orthogonalization]]; [[Iwasawa decomposition|Iwasawa decomposition]].
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:26, 6 June 2020


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.

Comments

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