Namespaces
Variants
Actions

Difference between revisions of "Orthogonalization"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
Line 13: Line 13:
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042019.png" />, are obtained from the condition of orthogonality of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042020.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042021.png" />. The geometric sense of this process comprises the fact that at every step, the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042022.png" /> is perpendicular to the linear hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042023.png" /> drawn to the end of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042024.png" />. The product of the lengths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042025.png" /> is equal to the volume of the parallelepiped constructed on the vectors of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042026.png" /> as edges. By normalizing the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042027.png" />, the required orthonormal system is obtained. An explicit expression of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042028.png" /> in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042029.png" /> is given by the formula
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042019.png" />, are obtained from the condition of orthogonality of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042020.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042021.png" />. The geometric sense of this process comprises the fact that at every step, the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042022.png" /> is perpendicular to the linear hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042023.png" /> drawn to the end of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042024.png" />. The product of the lengths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042025.png" /> is equal to the volume of the parallelepiped constructed on the vectors of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042026.png" /> as edges. By normalizing the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042027.png" />, the required orthonormal system is obtained. An explicit expression of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042028.png" /> in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042029.png" /> is given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042030.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042030.png" />/''G''<sub>''i''-1</sub></td> </tr></table>
  
(the determinant at the right-hand side has to be formally expanded by the last column). The corresponding orthonormal system takes the form
+
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042032.png" /> is the [[Gram determinant|Gram determinant]] of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042033.png" />, with ''G''<sub>0</sub>=1 by definition. (The determinant at the right-hand side has to be formally expanded by the last column).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042031.png" /></td> </tr></table>
+
The norm of these orthogonal vectors is given by ||''b''<sub>''i''</sub>||=SQRT(''G''<sub>''i''</sub>/''G''<sub>''i''-1</sub>). Thus, the corresponding orthonormal system takes the form
 +
 
 +
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042031.png" /> · ''G''<sub>''i''-1</sub> = ''b''<sub>''i''</sub> · SQRT(''G''<sub>''i''-1</sub> / ''G''<sub>''i''</sub>) </td> </tr></table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042032.png" /> is the [[Gram determinant|Gram determinant]] of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070420/o07042033.png" />.
 
  
 
This process can also be used for a countable system of vectors.
 
This process can also be used for a countable system of vectors.

Revision as of 09:48, 25 April 2016

orthogonalization process

An algorithm to construct for a given linear independent system of vectors in a Euclidean or Hermitian space an orthogonal system of non-zero vectors generating the same subspace in . The most well-known is the Schmidt (or Gram–Schmidt) orthogonalization process, in which from a linear independent system , an orthogonal system is constructed such that every vector () is linearly expressed in terms of , i.e. , where is an upper-triangular matrix. It is possible to construct the system such that it is orthonormal and such that the diagonal entries of are positive; the system and the matrix are defined uniquely by these conditions.

The Gram–Schmidt process is as follows. Put ; if the vectors have already been constructed, then

where

, are obtained from the condition of orthogonality of the vector to . The geometric sense of this process comprises the fact that at every step, the vector is perpendicular to the linear hull of drawn to the end of the vector . The product of the lengths is equal to the volume of the parallelepiped constructed on the vectors of the system as edges. By normalizing the vectors , the required orthonormal system is obtained. An explicit expression of the vectors in terms of is given by the formula

/Gi-1

where is the Gram determinant of the system , with G0=1 by definition. (The determinant at the right-hand side has to be formally expanded by the last column).

The norm of these orthogonal vectors is given by ||bi||=SQRT(Gi/Gi-1). Thus, the corresponding orthonormal system takes the form

· Gi-1 = bi · SQRT(Gi-1 / Gi)


This process can also be used for a countable system of vectors.

The Gram–Schmidt process can be interpreted as expansion of a non-singular square matrix in the product of an orthogonal (or unitary, in the case of a Hermitian space) and an upper-triangular matrix with positive diagonal entries, this product being a particular example of an Iwasawa decomposition.

References

[1] F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)
[2] A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)
How to Cite This Entry:
Orthogonalization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonalization&oldid=38640
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article